gale  'Bicentennial  publication? 

ELEMENTARY   PRINCIPLES   IN 
STATISTICAL   MECHANICS 


pale  bicentennial  publications 

With  the  approval  if  tbt  Prindent  and  FcUmn 
of  Tali  Unrveriity,  a  stria  of  volumes  has  keen 
prepared  by  a  number  of  the  Pnfesson  and  In- 
structorsj  to  be  issued  in  connection  with  the 
Bicentennial  Anniversary^  as  a  partial  indica- 
ttm  of  the  character  of  the  studies  in  wbicb  the 
University  teachers  are  engaged. 

This  series   of  volumes    is    respectfully  dedicated  4» 

ff)r  ^nurtures  of  tljr 


ELEMENTARY  PRINCIPLES 


IN 


STATISTICAL  MECHANICS 

DEVELOPED   WITH    ESPECIAL  REFERENCE  TO 

THE   RATIONAL   FOUNDATION   OF 
THERMODYNAMICS 


BY 

J.  WILLARD  GIBBS 

Proftuor  of  Matktmatual  Pkyrict  in  YaU  University 


OF   r 

UNIVERSITY 

OF 


NEW  YORK :   CHARLES  SCRIBNER'S  SONS 

LONDON:   EDWARD  ARNOLD 

1902 


A<> 
' 


Copyright,  1902, 
BY  CHARLES  SCRIBNER'S  SONS 

Published,   March,  zgoz. 


UNIVERSITY   PRESS    •    JOHN    WILSON 
AND     SON     •     CAMBRIDGE,    U.S.A. 


PREFACE. 

THE  usual  point  of  view  in  the  study  of  mechanics  is  that 
where  the  attention  is  mainly  directed  to  the  changes  which 
take  place  in  the  course  of  time  in  a  given  system.  The  prin- 
cipal problem  is  the  determination  of  the  condition  of  the 
system  with  respect  to.  configuration  and  velocities  at  any 
required  time,  when  its  condition  in  these  respects  has  been 
given  for  some  one  time,  and  the  fundamental  equations  are 
those  which  express  the  changes  continually  taking  place  in 
the  system.  Inquiries  of  this  kind  are  often  simplified  by 
taking  into  consideration  conditions  of  the  system  other  than 
those  through  which  it  actually  passes  or  is  supposed  to  pass, 
but  our  attention  is  not  usually  carried  beyond  conditions 
differing  infinitesimally  from  those  which  are  regarded  as 
actual. 

For  some  purposes,  however,  it  is  desirable  to  take  a  broader 
view  of  the  subject.  We  may  imagine  a  great  number  of 
systems  of  the  same  nature,  but  differing  in  the  configura- 
tions and  velocities  which  they  have  at  a  given  instant,  and 
differing  not  merely  infinitesimally,  but  it  may  be  so  as  to 
embrace  every  conceivable  combination  of  configuration  and 
velocities.  And  here  we  may  set  the  problem,  not  to  follow 
a  particular  system  through  its  succession  of  configurations, 
but  to  determine  how  the  whole  number  of  systems  will  be 
distributed  among  the  various  conceivable  configurations  and 
velocities  at  any  required  time,  when  the  distribution  has 
been  given  for  some  one  time.  The  fundamental  equation 
for  this  inquiry  is  that  which  gives  the  rate  of  change  of  the 
number  of  systems  which  fall  within  any  infinitesimal  limits 
of  configuration  and  velocity. 


94203 


viii  PREFACE. 

Such  inquiries  have  been  called  by  Maxwell  statistical. 
They  belong  to  a  branch  of  mechanics  which  owes  its  origin  to 
the  desire  to' explain  the  laws  of  thermodynamics  on  mechan- 
ical principles,  and  of  which  Clausius,  Maxwell,  and  Boltz- 
mann  are  to  be  regarded  as  the  principal  founders.  The  first 
inquiries  in  this  field  were  indeed  somewhat  narrower  in  their 
scope  than  that  which  has  been  mentioned,  being  applied  to 
the  particles  of  a  system,  rather  than  to  independent  systems. 
Statistical  inquiries  were  next  directed  to  the  phases  (or  con- 
ditions with  respect  to  configuration  and  velocity)  which 
succeed  one  another  in  a  given  system  in  the  course  of  time. 
The  explicit  consideration  of  a  great  number  of  systems  and 
their  distribution  in  phase,  and  of  the  permanence  or  alteration 
of  this  distribution  in  the  course  of  time  is  perhaps  first  found 
in  Boltzmann's  paper  on  the  "  Zusammenhang  zwischen  den 
Satzen  iiber  das  Verhalten  mehratomiger  Gasmolekiile  mit 
Jacobi's  Princip  des  letzten  Multiplicators  "  (1871). 

But  although,  as  a  matter  of  history,  statistical  mechanics 
owes  its  origin  to  investigations  in  thermodynamics,  it  seems 
eminently  worthy  of  an  independent  development,  both  on 
account  of  the  elegance  and  simplicity  of  its  principles,  and 
because  it  yields  new  results  and  places  old  truths  in  a  new 
light  in  departments  quite  outside  of  thermodynamics.  More- 
over, the  separate  study  of  this  branch  of  mechanics  seems  to 
afford  the  best  foundation  for  the  study  of  rational  thermody- 
namics and  molecular  mechanics. 

The  laws  of  thermodynamics,  as  empirically  determined, 
express  the  approximate  and  probable  behavior  of  systems  of 
a  great  number  of  particles,  or,  more  precisely,  they  express 
the  laws  of  mechanics  for  such  systems  as  they  appear  to 
beings  who  have  not  the  fineness  of  perception  to  enable 
them  to  appreciate  quantities  of  the  order  of  magnitude  of 
those  which  relate  to  single  particles,  and  who  cannot  repeat 
their  experiments  often  enough  to  obtain  any  but  the  most 
probable  results.  The  laws  of  statistical  mechanics  apply  to 
conservative  systems  of  any  number  of  degrees  of  freedom, 


PREFACE.  ix 

and  are  exact.  This  does  not  make  them  more  difficult  to 
establish  than  the  approximate  laws  for  systems  of  a  great 
many  degrees  of  freedom,  or  for  limited  classes  of  such 
systems.  The  reverse  is  rather  the  case,  for  our  attention  is 
not  diverted  from  what  is  essential  by  the  peculiarities  of  the 
system  considered,  and  we  are  not  obliged  to  satisfy  ourselves 
that  the  effect  of  the  quantities  and  circumstances  neglected 
will  be  negligible  in  the  result.  The  laws  of  thermodynamics 
may  be  easily  obtained  from  the  principles  of  statistical  me- 
chanics, of  which  they  are  the  incomplete  expression,  but 
they  make  a  somewhat  blind  guide  in  our  search  for  those 
laws.  This  is  perhaps  the  principal  cause  of  the  slow  progress 
of  rational  thermodynamics,  as  contrasted  with  the  rapid  de- 
duction of  the  consequences  of  its  laws  as  empirically  estab- 
lished. To  this  must  be  added  that  the  rational  foundation 
of  thermodynamics  lay  in  a  branch  of  mechanics  of  which 
the  fundamental  notions  and  principles,  and  the  characteristic 
operations,  were  alike  unfamiliar  to  students  of  mechanics. 

We  may  therefore  confidently  believe  that  nothing  will 
more  conduce  to  the  clear  apprehension  of  the  relation  of 
thermodynamics  to  rational  mechanics,  and  to  the  interpreta- 
tion of  observed  phenomena  with  reference  to  their  evidence 
respecting  the  molecular  constitution  of  bodies,  than  the 
study  of  the  fundamental  notions  and  principles  of  that  de- 
partment of  mechanics  to  which  thermodynamics  is  especially 
related. 

Moreover,  we  avoid  the  gravest  difficulties  when,  giving  up 
the  attempt  to  frame  hypotheses  concerning  the  constitution 
of  material  bodies,  we  pursue  statistical  inquiries  as  a  branch 
of  rational  mechanics.  In  the  present  state  of  science,  it 
seems  hardly  possible  to  frame  a  dynamic  theory  of  molecular 
action  which  shall  embrace  the  phenomena  of  thermody- 
namics, of  radiation,  and  of  the  electrical  manifestations 
which  accompany  the  union  of  atoms.  Yet  any  theory  is 
obviously  inadequate  which  does  not  take  account  of  all 
these  phenomena.  Even  if  we  confine  cur  attention  to  the 


X  PREFACE. 

phenomena  distinctively  thermodynamic,  we  do  not  escape 
difficulties  in  as  simple  a  matter  as  the  number  of  degrees 
of  freedom  of  a  diatomic  gas.  It  is  well  known  that  while 
theory  would  assign  to  the  gas  six  degrees  of  freedom  per 
molecule,  in  our  experiments  on  specific  heat  we  cannot  ac- 
count for  more  than  five.  Certainly,  one  is  building  on  an 
insecure  foundation,  who  rests  his  work  on  hypotheses  con- 
cerning the  constitution  of  matter. 

Difficulties  of  this  kind  have  deterred  the  author  from  at- 
tempting to  explain  the  mysteries  of  nature,  and  have  forced 
him  to  be  contented  with  the  more  modest  aim  of  deducing 
some  of  the  more  obvious  propositions  relating  to  the  statis- 
tical branch  of  mechanics.  Here,  there  can  be  no  mistake  in 
regard  to  the  agreement  of  the  hypotheses  with  the  facts  of 
nature,  for  nothing  is  assumed  in  that  respect.  The  only 
error  into  which  one  can  fall,  is  the  want  of  agreement  be- 
tween the  premises  and  the  conclusions,  and  this,  with  care, 
one  may  hope,  in  the  main,  to  avoid. 

The  matter  of  the  present  volume  consists  in  large  measure 
of  results  which  have  been  obtained  by  the  investigators 
mentioned  above,  although  the  point  of  view  and  the  arrange- 
ment may  be  different.  These  results,  given  to  the  public 
one  by  one  in  the  order  of  their  discovery,  have  necessarily, 
in  their  original  presentation,  not  been  arranged  in  the  most 
logical  manner. 

In  the  first  chapter  we  consider  the  general  problem  which 
has  been  mentioned,  and  find  what  may  be  called  the  funda- 
mental equation  of  statistical  mechanics.  A  particular  case 
of  this  equation  will  give  the  condition  of  statistical  equi- 
librium, i.  e.,  the  condition  which  the  distribution  of  the 
systems  in  phase  must  satisfy  in  order  that  the  distribution 
shall  be  permanent.  In  the  general  case,  the  fundamental 
equation  admits  an  integration,  which  gives  a  principle  which 
may  be  variously  expressed,  according  to  the  point  of  view 
from  which  it  is  regarded,  as  the  conservation  of  density-in- 
phase,  or  of  extension-in-phase,  or  of  probability  of  phase. 


PREFACE.  xi 

In  the  second  chapter,  we  apply  this  principle  of  conserva- 
tion of  probability  of  phase  to  the  theory  of  errors  in  the 
calculated  phases  of  a  system,  when  the  determination  of  the 
arbitrary  constants  of  the  integral  equations  are  subject  to 
error.  In  this  application,  we  do  not  go  beyond  the  usual 
approximations.  In  other  words,  we  combine  the  principle 
of  conservation  of  probability  of  phase,  which  is  exact,  with 
those  approximate  relations,  which  it  is  customary  to  assume 
in  the  "  theory  of  errors." 

In  the  third  chapter  we  apply  the  principle  of  conservation 
of  extension-in-phase  to  the  integration  of  the  differential 
equations  of  motion.  This  gives  Jacobi's  "  last  multiplier," 
as  has  been  shown  by  Boltzmann. 

In  the  fourth  and  following  chapters  we  return  to  the  con- 
sideration of  statistical  equilibrium,  and  confine  our  attention 
to  conservative  systems.  We  consider  especially  ensembles 
of  systems  in  which  the  index  (or  logarithm)  of  probability  of 
phase  is  a  linear  function  of  the  energy.  This  distribution, 
on  account  of  its  unique  importance  in  the  theory  of  statisti- 
cal equilibrium,  I  have  ventured  to  call  canonical,  and  the 
divisor  of  the  energy,  the  modulus  of  distribution.  The 
moduli  of  ensembles  have  properties  analogous  to  temperature, 
in  that  equality  of  the  moduli  is  a  condition  of  equilibrium 
with  respect  to  exchange  of  energy,  when  such  exchange  is 
made  possible. 

We  find  a  differential  equation  relating  to  average  values 
in  the  ensemble  which  is  identical  in  form  with  the  funda- 
mental differential  equation  of  thermodynamics,  the  average 
index  of  probability  of  phase,  with  change  of  sign,  correspond- 
ing to  entropy,  and  the  modulus  to  temperature. 

For  the  average  square  of  the  anomalies  of  the  energy,  we 
find  an  expression  which  vanishes  in  comparison  with  the 
square  of  the  average  energy,  when  the  number  of  degrees 
of  freedom  is  indefinitely  increased.  An  ensemble  of  systems 
in  which  the  number  of  degrees  of  freedom  is  of  the  same 
order  of  magnitude  as  the  number  of  molecules  in  the  bodies 


xii  PREFACE. 

with  which  we  experiment,  if  distributed  canonically,  would 
therefore  appear  to  human  observation  as  an  ensemble  of 
systems  in  which  all  have  the  same  energy. 

We  meet  with  other  quantities,  in  the  development  of  the 
subject,  which,  when  the  number  of  degrees  of  freedom  is 
very  great,  coincide  sensibly  with  the  modulus,  and  with  the 
average  index  of  probability,  taken  negatively,  in  a  canonical 
ensemble,  and  which,  therefore,  may  also  be  regarded  as  cor- 
responding to  temperature  and  entropy.  The  correspondence 
is  however  imperfect,  when  the  number  of  degrees  of  freedom 
is  not  very  great,  and  there  is  nothing  to  recommend  these 
quantities  except  that  in  definition  they  may  be  regarded  as 
more  simple  than  those  which  have  been  mentioned.  In 
Chapter  XIV,  this  subject  of  thermodynamic  analogies  is 
discussed  somewhat  at  length. 

Finally,  in  Chapter  XV,  we  consider  the  modification  of 
the  preceding  results  which  is  necessary  when  we  consider 
systems  composed  of  a  number  of  entirely  similar  particles, 
or,  it  may  be,  of  a  number  of  particles  of  several  kinds,  all  of 
each  kind  being  entirely  similar  to  each  other,  and  when  one 
of  the  variations  to  be  considered  is  that  of  the  numbers  of 
the  particles  of  the  various  kinds  which  are  contained  in  a 
system.  This  supposition  would  naturally  have  been  intro- 
duced earlier,  if  our  object  had  been  simply  the  expression  of 
the  laws  of  nature.  It  seemed  desirable,  however,  to  separate 
sharply  the  purely  thermodynamic  laws  from  those  special 
modifications  which  belong  rather  to  the  theoiy  of  the  prop- 
erties of  matter. 

J.  W.  G. 

NEW  HAVEN,  December,  1901. 


CONTENTS. 


CHAPTER  I. 

GENERAL    NOTIONS.      THE    PRINCIPLE    OF    CONSERVATION 

OF  EXTENSION-IN-PHASE. 

PAGE 

Hamilton's  equations  of  motion 3-5 

Ensemble  of  systems  distributed  in  phase 5 

Extension-in-phase,  density-in-phase 6 

Fundamental  equation  of  statistical  mechanics 6-8 

Condition  of  statistical  equilibrium 8 

Principle  of  conservation  of  density-in-phase 9 

Principle  of  conservation  of  extension-in-phase 10 

Analogy  in  hydrodynamics 11 

Extension-in-phase  is  an  invariant 11-13 

Dimensions  of  extension-in-phase 13 

Various  analytical  expressions  of  the  principle 13-15 

Coefficient  and  index  of  probability  of  phase 16 

Principle  of  conservation  of  probability  of  phase 17,  18 

Dimensions  of  coefficient  of  probability  of  phase 19 


CHAPTER  II. 

APPLICATION  OF    THE    PRINCIPLE   OF    CONSERVATION    OF 
EXTENSION-IN-PHASE  TO  THE   THEORY  OF  ERRORS. 

Approximate  expression  for  the  index  of  probability  of  phase    .    20,  21 
Application  of  the  principle  of  conservation  of  probability  of  phase 
to  the  constants  of  this  expression 21-25 


CHAPTER  III. 

APPLICATION  OF  THE  PRINCIPLE  OF  CONSERVATION  OF 
EXTENSION-IN-PHASE  TO  THE  INTEGRATION  OF  THE 
DIFFERENTIAL  EQUATIONS  OF  MOTION. 

Case  in  which  the  forces  are  function  of  the  coordinates  alone     .    26-29 
Case  in  which  the  forces  are  functions  of  the  coordinates  with  the 
time 30,  31 


xiv  CONTENTS. 


CHAPTER   IV. 

ON  THE  DISTRIBUTION-IN-PHASE  CALLED  CANONICAL,  IN 
WHICH   THE    INDEX    OF    PROBABILITY    IS    A   LINEAR 

FUNCTION   OF  THE  ENERGY. 

PAGE 

Condition  of  statistical  equilibrium 32 

Other  conditions  which  the  coefficient  of  probability  must  satisfy  .  33 

""""  Canonical  distribution  —  Modulus  of  distribution 34 

^  must  be  finite 35 

The  modulus  of  the  canonical  distribution  has  properties  analogous 

to  temperature 35-37 

Other  distributions  have  similar  properties 37 

Distribution  in  which  the  index  of  probability  is  a  linear  function  of 

the  energy  and  of  the  moments  of  momentum  about  three  axes  .  38,  39 
Case  in  which  the  forces  are  linear  functions  of  the  displacements, 

and  the  index  is  a.  linear  function  of  the  separate  energies  relating 

to  the  normal  types  of  motion 39-41 

Differential  equation  relating  to  average  values  in  a  canonical 

ensemble 42-44 

This  is  identical  in  form  with  the  fundamental  differential  equation 

of  thermodynamics 44,  45 

CHAPTER  V. 

AVERAGE  VALUES  IN  A  CANONICAL   ENSEMBLE    OF    SYS- 
TEMS. 
Case  of  v  material  points.     Average  value  of  kinetic  energy  of  a 

single  point  for  a  given  configuration  or  for  the  whole  ensemble 

=  f  0 46,  47 

Average  value  of  total  kinetic  energy  for  any  given  configuration 

or  for  the  whole  ensemble  =  %  v  0 47 

System  of  n  degrees  of  freedom.     Average  value  of  kinetic  energy, 

for  any  given  configuration  or  for  the  whole  ensemble  =  f  0  .    48-50 

Second  proof  of  the  same  proposition 50-52 

Distribution  of  canonical  ensemble  in  configuration 52-54 

Ensembles  canonically  distributed  in  configuration 55 

Ensembles  canonically  distributed  in  velocity 56 

CHAPTER  VI. 

EXTENSION1-IN-CONFIGURATION       AND      EXTENSION-TN- 
VELOCITY. 

Extension-in-configuration    and    extension-in-velocity    are     invari- 
ants .    57-59 


CONTENTS.  XV 

PAGE 

Dimensions  of  these  quantities 60 

Index  and  coefficient  of  probability  of  configuration 61 

Index  and  coefficient  of  probability  of  velocity 62 

Dimensions  of  these  coefficients 63 

Relation  between  extension-in-configuration  and  extension-in-velocity     64 
Definitions  of  extension-in-phase,  extension-in-configuration,  and  ex- 
tension-in- velocity,  without  explicit  mention  of  coordinates  .     .    65-67 


CHAPTER  VII. 

FARTHER    DISCUSSION    OF   AVERAGES    IN    A    CANONICAL 
ENSEMBLE  OF   SYSTEMS. 

Second  and  third  differential  equations  relating  to  average  values 

in  a  canonical  ensemble 68,  69 

These  are  identical  in  form  with  thermodynamic  equations  enun- 
ciated by  Clausius 69 

Average  square  of  the  anomaly  of  the  energy  —  of  the  kinetic  en- 
ergy—  of  the  potential  energy 70-72 

These  anomalies  are  insensible  to  human  observation  and  experi- 
ence when  the  number  of  degrees  of  freedom  of  the  system  is  very 

great 73,  74 

Average  values  of  powers  of  the  energies 75-77 

Average  values  of  powers  of  the  anomalies  of  the  energies  .  .  77-80 
Average  values  relating  to  forces  exerted  on  external  bodies  .  .  80-83 
General  formulae  relating  to  averages  in  a  canonical  ensemble  .  83-86 


CHAPTER  VIII. 

ON  CERTAIN  IMPORTANT    FUNCTIONS  OF    THE  ENERGIES 
OF  A   SYSTEM. 

Definitions.     V  =  extension-in-phase  below  a  limiting  energy  (e). 

$  =  \o«dVldc 87,88 

Vq  =  extension-in-configuration  below  a  limiting  value  of  the  poten- 
tial energy  (e?).     fa  =  \o^dVqjdfq 89,90 

Vp  =  extension-in-velocity  below  a  limiting  value  of  the  kinetic  energy 

(*).     ^p  =  loSdVpjd€p 90,91 

Evaluation  of  Vp  and  $p 91-93 

Average  values  of  functions  of  the  kinetic  energy 94,  95 

Calculation  of   FfromF^ 95,96 

Approximate  formulae  for  large  values  of  n 97,98 

Calculation  of  V  or  <£  for  whole  system  when  given  for  parts  ...     98 
Geometrical  illustration  .  99 


xvi  CONTENTS. 

CHAPTER  IX. 

THE  FUNCTION  </>  AND  THE  CANONICAL  DISTRIBUTION. 

When  n  >  2,  the  most  probable  value  of  the  energy  in  a  canonical 
ensemble  is  determined  by  d(j>  j  de  =  1  /  e 100,101 

When  n  >  2,  the  average  value  of  d$  j  de  in  a  canonical  ensemble 
isl/e 101 

When  n  is  large,  the  value  of  <£  corresponding  to  d(f>/de=l/Q 
(<£o)  js  nearly  equivalent  (except  for  an  additive  constant)  to 
the  average  index  of  probability  taken  negatively  (—  fj)  .  .  101-104 

Approximate  formulae  for  <£0  +  fj  when  n  is  large 104-106 

When  n  is  large,  the  distribution  of  a  canonical  ensemble  in  energy 
follows  approximately  the  law  of  errors 105 

This  is  not  peculiar  to  the  canonical  distribution 107,  108 

Averages  in  a  canonical  ensemble 108-114 

CHAPTER  X. 

ON  A  DISTRIBUTION  IN  PHASE  CALLED  MICROCANONI- 
CAL IN  WHICH  ALL  THE  SYSTEMS  HAVE  THE  SAME 
ENERGY. 

The  microcanonical  distribution  denned  as  the  limiting  distribution 
obtained  by  various  processes 115,  116 

Average  values  in  the  microcanonical  ensemble  of  functions  of  the 
kinetic  and  potential  energies 117-120 

If  two  quantities  have  the  same  average  values  in  every  microcanon- 
ical ensemble,  they  have  the  same  average  value  in  every  canon- 
ical ensemble 120 

Average  values  in  the  microcanonical  ensemble  of  functions  of  the 
energies  of  parts  of  the  system 121-123 

Average  values  of  functions  of  the  kinetic  energy  of  a  part  of  the 
system 123,  124 

Average  values  of  the  external  forces  in  a  microcanonical  ensemble. 
Differential  equation  relating  to  these  averages,  having  the  form 
of  the  fundamental  differential  equation  of  thermodynamics  .  124-128 

CHAPTER  XI. 

MAXIMUM  AND  MINIMUM  PROPERTIES  OF  VARIOUS  DIS- 
TRIBUTIONS IN  PHASE. 

Theorems  I- VI.     Minimum  properties  of  certain  distributions  .  129-133 
Theorem  VII.     The  average  index  of  the  whole  system  compared 
with  the  sum  of  the  average  indices  of  the  parts 133-135 


CONTENTS.  xvii 

PAGE 

Theorem  VIII.     The  average  index  of  the  whole  ensemble  com- 
pared with  the  average  indices  of  parts  of  the  ensemble      .     .  135-137 
Theorem  IX.     Effect  on  the  average  index  of  making  the  distribu- 
tion-in-phase  uniform  within  any  limits 137-138 

CHAPTER  XII. 

ON  THE    MOTION  OF  SYSTEMS     AND  ENSEMBLES    OF  SYS- 
TEMS THROUGH  LONG  PERIODS  OF  TIME. 

Under  what  conditions,  and  with  what  limitations,  may  we  assume 
that  a  system  will  return  in  the  course  of  time  to  its  original 
phase,  at  least  to  any  required  degree  of  approximation?  .  .  139-142 

Tendency  in  an  ensemble  of  isolated  systems  toward  a  state  of  sta- 
tistical equilibrium 143-151 

CHAPTER  XIII. 

EFFECT    OF    VARIOUS    PROCESSES    ON    AN   ENSEMBLE    OF 
SYSTEMS. 

Variation  of  the  external  coordinates  can  only  cause  a  decrease  in 
the  average  index  of  probability 152-154 

This  decrease  may  in  general  be  diminished  by  diminishing  the 
rapidity  of  the  change  in  the  external  coordinates  ....  154-157 

The  mutual  action  of  two  ensembles  can  only  diminish  the  sum  of 
their  average  indices  of  probability 158, 159 

In  the  mutual  action  of  two  ensembles  which  are  canonically  dis- 
tributed, that  which  has  the  greater  modulus  will  lose  energy  .  160 

Repeated  action  between  any  ensemble  and  others  which  are  canon- 
ically distributed  with  the  same  modulus  will  tend  to  distribute 
the  first-mentioned  ensemble  canonically  with  the  same  modulus  161 

Process  analogous  to  a  Carnot's  cycle 162,163 

Analogous  processes  in  thermodynamics 163, 164 

CHAPTER  XIV. 

DISCUSSION   OF  THERMODYNAMIC  ANALOGIES. 

The  finding  in  rational  mechanics  an  a  priori  foundation  forthermo- 
dynamics  requires  mechanical  definitions  of  temperature  and 
entropy.  Conditions  which  the  quantities  thus  defined  must 
satisfy 165-167 

The  modulus  of  a  canonical  ensemble  (0),  and  the  average  index  of 
probability  taken  negatively  (rj),  as  analogues  of  temperature 
and  entropy 167-169 


xviii  CONTENTS. 

PAGE 

The  functions  of  the  energy  del d  log  Fand  log  Fas  analogues  of 

temperature  and  entropy 169-172 

The  functions  of  the  energy  de  /  cty  and  <p  as  analogues  of  tempera- 
ture and  entropy 1 72-1 78 

Merits  of  the  different  systems 178-183 

If  a  system  of  a  great  number  of  degrees  of  freedom  is  microcanon- 
ically  distributed  in  phase,  any  very  small  part  of  it  may  be  re- 
garded as  canonically  distributed 183 

Units  of  0  and  rj  compared  with  those  of  temperature  and 
entropy 183-186 

CHAPTER  XV. 

SYSTEMS  COMPOSED  OF  MOLECULES. 

Generic  and  specific  definitions  of  a  phase 187-189 

Statistical  equilibrium  with  respect  to  phases  generically  defined 

and  with  respect  to  phases  specifically  defined 189 

Grand  ensembles,  petit  ensembles 189,190 

Grand  ensemble  canonically  distributed 190-193 

Q  must  be  finite 193 

Equilibrium  with  respect  to  gain  or  loss  of  molecules    ....  194-197 
Average  value  of  any  quantity  in  grand  ensemble  canonically  dis- 
tributed   198 

Differential  equation  identical  in  form  with  fundamental  differen- 
tial equation  in  thermodynamics 199,  200 

Average  value  of  number  of  any  kind  of  molecules  (i>)     .     .     .     .     201 

Average  value  of  (v-v)* 201,202 

Comparison  of  indices 203-206 

When  the  number  of  particles  in  a  system  is  to  be  treated  as 
variable,  the  average  index  of  probability  for  phases  generically 
defined  corresponds  to  entropy 206 


ELEMENTARY  PRINCIPLES    IN 
STATISTICAL  MECHANICS 


((  UNIVERSITY  J 


ELEMENTARY    PRINCIPLES    IN 
STATISTICAL  MECHANICS 


CHAPTER  I. 

GENERAL  NOTIONS.    THE   PRINCIPLE  OF 
OF  EXTENSION-IN-PHASE. 

WE  shall  use  Hamilton's  form  of  the  equations  of  motion  for 
a  system  of  n  degrees  of  freedom,  writing  ql ,  . .  ,qn  for  the 
(generalized)  coordinates,  qi ,  . . .  qn  for  the  (generalized)  ve- 
locities, and 

for  the  moment  of  the  forces.  We  shall  call  the  quantities 
Fl9...Fn  the  (generalized)  forces,  and  the  quantities  p1 . . .  pn, 
defined  by  the  equations 

Pl  =  ^-t    p2  =  ^,    etc.,  (2) 

dqi  dq2 

where  ep  denotes  the  kinetic  energy  of  the  system,  the  (gen- 
eralized) momenta.  The  kinetic  energy  is  here  regarded  as 
a  function  of  the  velocities  and  coordinates.  We  shall  usually 
regard  it  as  a  function  of  the  momenta  and  coordinates,* 
and  on  this  account  we  denote  it  by  ep.  This  will  not  pre- 
vent us  from  occasionally  using  formulae  like  (2),  where  it  is 
sufficiently  evident  the  kinetic  energy  is  regarded  as  function 
of  the  g's  and  ^'s.  But  in  expressions  like  dep/dq1 ,  where  the 
denominator  does  not  determine  the  question,  the  kinetic 

*  The  use  of  the  momenta  instead  of  the  velocities  as  independent  variables 
is  the  characteristic  of  Hamilton's  method  which  gives  his  equations  of  motion 
their  remarkable  degree  of  simplicity.  We  shall  find  that  the  fundamental 
notions  of  statistical  mechanics  are  most  easily  defined,  and  are  expressed  in 
the  most  simple  form,  when  the  momenta  with  the  coordinates  are  used  to 
describe  the  state  of  a  system. 


4  HAMILTON'S  EQUATIONS. 

energy  is  always  to  be  treated  in  the  differentiation  as  function 
of  the  p's  and  q*s. 
We  have  then 

*  =  ;fe*  *l  =  -^  +  Fl'  etc>  (3) 

These  equations  will  hold  for  any  forces  whatever.  If  the 
'fetces^  &i*e  £  dptt§erVative,  in  other  words,  if  the  expression  (1) 
j.stant  exact  differential,  we  may  set 


where  eq  is  a  function  of  the  coordinates  which  we  shall  call 
the  potential  energy  of  the  system.  If  we  write  e  for  the 
total  energy,  we  shall  have 

e  =  €P  +  e«>  (5) 

and  equatipns  (3)  may  be  written 

*'  =  ;£'  *  =  -£'  etc-  [I          <«> 

The  potential  energy  (e3)  may  depend  on  other  variables 
beside  the  coordinates  q1  .  .  .  qn.  We  shall  often  suppose  it  to 
depend  in  part  on  coordinates  of  external  bodies,  which  we 
shall  denote  by  ax  ,  #2  ,  etc.  We  shall  then  have  for  the  com- 
plete value  of  the  differential  of  the  potential  energy  * 

deq  =  —  FI  dql  .  .  —  Fn  dqn  —  A1  da^  —  A2  daz  —  etc.,          (7) 

where  A^  A%,  etc.,  represent  forces  (in  the  generalized  sense) 
exerted  by  the  system  on  external  bodies.  For  the  total  energy 
(e)  we  shall  have 

de=qldpl  .  .  .  +  qndpn~Pidqi  .  .  . 

—  pn  dqn  —  Al  da-i  —  A2  daz  —  etc.  (8) 

It  will  be  observed  that  the  kinetic  energy  (e^,)  in  the 
most  general  case  is  a  quadratic  function  of  the  p's  (or  g-'s) 

*  It  will  be  observed,  that  although  we  call  e  the  potential  energy  of  the 
system  which  we  are  considering,  it  is  really  so  defined  as  to  include  that 
energy  which  might  be  described  as  mutual  to  that  system  and  external 
bodies. 


ENSEMBLE   OF  SYSTEMS.  5 

v 

involving  also  the  ^'s  but  not  the  a's  ;  that  the  potential  energy, 
when  it  exists,  is  function  of  the  <?'s  and  a's  ;  and  that  the 
total  energy,  when  it  exists,  is  function  of  the  jt?'s  (or  ^s),  the 
9's,  and  the  a's.  In  expressions  like  dejdq^  them's,  and  not 
the  q's,  are  to  be  taken  as  independent  variables,  as  has  already 
been  stated  with  respect  to  the  kinetic  energy. 

Lev  us  imagine  a  great  number  of  independent  systems, 
identical  in  nature,  but  differing  in  phase,  that  is,  in  their 
condition  with  respect  to  configuration  and  velocity.  The 
forces  are  supposed  to  be  determined  for  every  system  by  the 
same  law,  being  functions  of  the  coordinates  of  the  system 
q19  .  .  .  qn,  either  alone  or  with  the  coordinates  a1?  a2,  etc.  of 
certain  external  bodies.  It  is  not  necessary  that  they  should 
be  derivable  from  a  force-function.  The  external  coordinates 
a15  a2,  etc.  may  vary  with  the  time,  but  at  any  given  time 
have  fixed  values.  In  this  they  differ  from  the  internal 
coordinates  q1  ,  .  .  .  qn  ,  which  at  the  same  time  have  different 
values  in  the  different  systems  considered. 

Let  us  especially  consider  the  number  of  systems  which  at  a 
given  instant  fall  within  specified  limits  of  phase,  viz.,  those 
for  which 

Pi   <Pi<  Pi",  qi  <qi<  q", 


Pn    <Pn<  P",  qn    <  q»  < 

the  accented  letters  denoting  constants.  We  shall  suppose 
the  differences  p^'  —  p{,  q^  —  q^,  etc.  to  be  infinitesimal,  and 
that  the  systems  are  distributed  in  phase  in  some  continuous 
manner,*  so  that  the  number  having  phases  within  the  limits 
specified  may  be  represented  by 

i')  •  •  •  (?»"  -  ?„'),     (10) 


*  In  strictness,  a  finite  number  of  systems  cannot  be  distributed  contin- 
uously in  phase.  But  by  increasing  indefinitely  the  number  of  systems,  we 
may  approximate  to  a  continuous  law  of  distribution,  such  as  is  here 
described.  To  avoid  tedious  circumlocution,  language  like  the  above  may 
be  allowed,  although  wanting  in  precision  of  expression,  when  the  sense  in 
which  it  is  to  be  taken  appears  sufficiently  clear. 


6  VARIATION  OF  THE 

or  more  briefly  by 


.  .  .  dpn  dql  .  .  .  dqn,  (li) 

where  D  is  a  function  of  the  p's  and  q's  and  in  general  of  t  alb  3, 

for  as  time  goes  on,  and  the  individual  systems  change  the\r 

phases,  the  distribution  of  the  ensemble  in  phase  will  in  gen- 

eral vary.     In  special  cases,  the  distribution  in  phase  will 

remain  unchanged.     These  are  cases  of  statistical  equilibr  turn. 

If  we  regard  all  possible  phases  as  forming  a  sort  oi  exten- 

ision  of  2  n  dimensions,  we  may  regard  the  product  of  differ- 

fentials  in  (11)  as  expressing  an  element  of  this  extension,  and 

\D  as  expressing  the  density  of  the  systems  in  that  element. 

We  shall  call  the  product 

dpl...  dpn  dqlf.  .  dqn  (12) 

an  element  of  extensionrin-phase,  and  D  the  density-inr-phase 
of  the  systems. 

It  is  evident  that  the  changes  which  take  place  in  the  den- 
sity of  the  systems  in  any  given  element  of  extension-in- 
phase  will  depend  on"  the  dynamical  nature  of  the  systems 
and  their  distribution  in  phase  at  the  time  considered. 

In  the  case  of  conservative  systems,  with  which  we  shall  be 
principally  concerned,  their  dynamical  nature  is  completely 
determined  by  the  function  which  expresses  the  energy  (e)  in 
terms  of  the  |?'s,  <?'s,  and  a's  (a  function  supposed  identical 
for  all  the  systems)  ;  in  the  more  general  case  which  we  are 
considering,  the  dynamical  nature  of  the  systems  is  deter- 
mined by  the  functions  which  express  the  kinetic  energy  (ep) 
in  terms  of  the  p's  and  <?'s,  and  the  forces  in  terms  of  the 
<?'s  and  «'s.  The  distribution  in  phase  is  expressed  for  the 
time  considered  by  D  as  function  of  the  p's  and  q's.  To  find 
the  value  of  dD/dt  for  the  specified  element  of  extension-in- 
phase,  we  observe  that  the  number  of  systems  within  the 
limits  can  only  be  varied  by  systems  passing  the  limits,  which 
may  take  place  in  4  n  different  ways,  viz.,  by  the  pl  of  a  sys- 
tem passing  the  limit  p^,  or  the  limit  p/',  or  by  the  ql  of  a 
system  passing  the  limit  q^  or  the  limit  <?/',  etc.  Let  us 
consider  these  cases  separately. 


DENSITY-IN-PHASE.  1 

In  the  first  place,  let  us  consider  the  number  of  systems 
which  in  the  time  dt  pass  into  or  out  of  the  specified  element 
by  pl  passing  the  limit  p^.  It  will  be  convenient,  and  it  is 
evidently  allowable,  to  suppose  dt  so  small  that  the  quantities 
^  dt,  ql  dt,  etc.,  which  represent  the  increments  of  pl,  ql,  etc., 
in  the  time  dt  shall  be  infinitely  small  in  comparison  with 
the  infinitesimal  differences  p£  —  p^,  q±r  —  <?/,  etc.,  which  de- 
termine the  magnitude  of  the  element  of  extension-in-phase. 
The  systems  for  which  pl  passes  the  limit  p^  in  the  interval 
dt  are  those  for  which  at  the  commencement  of  this  interval 
the  value  of  p1  lies  between  p^  and  p^  —  p±  dt,  as  is  evident 
if  we  consider  separately  the  cases  in  which  pl  is  positive  and 
negative.  Those  systems  for  which  p1  lies  between  these 
limits,  and  the  other  p's  and  j's  between  the  limits  specified  in 
(9),  will  therefore  pass  into  or  out  of  the  element  considered 
according  aH^t?  is  positive  or  negative,  unless  indeed  they  also 
pass  some  other  limit  specified  in  (9)  during  the  same  inter- 

^val  of  time.  But  the  number  which  pass  any  two  of  these 
limits  will  be  represented  by  an  expression  containing  the 
square  of  dt  as  a  factor,  and  is  evidently  negligible,  when  dt 

1  is  sufficiently  small,  compared  with  the  number  which  we  are 
seeking  to  evaluate,  and  which  (with  neglect  of  terms  contain- 
ing dt2)  may  be  found  by  substituting  pl  dt  for  p^'  —  p^  in 
(10)  or  for  dp1  in  (11). 
The  expression 

Dpi  dt  dpz  .  .  .  dpn  dqi  .  .  .  dqn  (13) 

will  therefore  represent,  according  as  it  is  positive  or  negative, 
the  increase  or  decrease  of  the  number  of  systems  within  the 
given  limits  which  is  due  to  systems  passing  the  limit  p^.  A 
similar  expression,  in  which  however  D  and  p  will  have 
slightly  different  values  (being  determined  for  p^'  instead  of 
Pi),  will  represent  the  decrease  or  increase  of  the  number  of 
systems  due  to  the  passing  of  the  limit  p^'.  The  difference 
of  the  two  expressions,  or 


dpi  .  .  .  dpn  dqi  .  .  .  dqn  dt  (14) 


8  CONSERVATION  OF 

will  represent  algebraically  the  decrease  of  the  number  of 
systems  within  the  limits  due  to  systems  passing  the  limits  p^ 
and  PI'. 

The  decrease  in  the  number  of  systems  within  the  limits 
due  to  systems  passing  the  limits  q±  and  <?/'  may  be  found  in 
the  same  way.  This  will  give 


for  the  decrease  due  to  passing  the  four  limits  p±,  p^",  <?/,  q^'. 
But  since  the  equations  of  motion  (3)  give 

^  +  ^  =  0,  (16) 

dpi       dql 

the  expression  reduces  to 

(dD  •        dD  •  \ 
d^pi  +  d^  ?i)  *•  •  •  •  *•  dyi  •  •  •  *-•*•       (17) 

If  we  prefix  2  to  denote  summation  relative  to  the  suffixes 
1  ...  n,  we  get  the  total  decrease  in  the  number  of  systems 
within  the  limits  in  the  time  dt.  That  is, 

T~  i* 

-dDdPl...  dpn  dSl...  dqn,      (18) 


d~  ^  )  dpl  '  '  '  d    d    '  "  d    dt  ~ 


or 

where  the  suffix  applied  to  the  differential  coefficient  indicates 
that  the  JP'S  and  <?'s  are  to  be  regarded  as  constant  in  the  differ- 
entiation. The  condition  of  statistical  equilibrium  is  therefore 


If  at  any  instant  this  condition  is  fulfilled  for  all  values  of  the 
p's  and  <?'s,  (dD/dt}ptg  vanishes,  and  therefore  the  condition 
will  continue  to  hold,  and  the  distribution  in  phase  will  be 
permanent,  so  long  as  the  external  coordinates  remain  constant. 
But  the  statistical  equilibrium  would  in  general  be  disturbed 
by  a  change  in  the  values  of  the  external  coordinates,  which 


DENSITY-IN-PHASE.  9 

would  alter  the  values  of  tlie  jt?'s  as  determined  by  equations 
(3),  and  thus  disturb  the  relation  expressed  in  the  last  equation. 
If  we  write  equation  (19)  in  the  form 


it  will  be  seen  to  express  a  theorem  of  remarkable  simplicity. 
Since  D  is  a  function  of  t,  pl,  .  .  .  pn,  ql  ,  .  .  .  qn,  its  complete 
differential  will  consist  of  parts  due  to  the  variations  of  all 
these  quantities.  Now  the  first  term  of  the  equation  repre- 
sents the  increment  of  D  due  to  an  increment  of  t  (with  con- 
stant values  of  them's  and  ^'s),  and  the  rest  of  the  first  member 
represents  the  increments  of  D  due  to  increments  of  the  p's 
and  g's,  expressed  by  pl  dt,  ql  dt,  etc.  But  these  are  precisely 
the  increments  which  the  jt?'s  and  #'s  receive  in  the  movement 
of  a  system  in  the  tune  dt.  The  whole  expression  represents 
the  total  increment  of  D  for  the  varying  phase  of  a  moving 
system.  We  have  therefore  the  theorem  :  — 

In  an  ensemble  of  mechanical  systems  identical  in  nature  and 
subject  to  forces  determined  by  identical  laws,  but  distributed 
in  phase  in  any  continuous  manner,  the  density-in-phase  is 
constant  in  time  for  the  varying  phases  of  a  moving  system  ; 
provided,  that  the  forces  of  a  system  are  functions  of  its  co- 
ordinates, either  alone  or  with  the  time.* 

This  may  be  called  the  principle  of  conservation  of  density- 
in-phase.  It  may  also  be  written 

(fL.,=»- 


where  a, . . .  h  represent  the  arbitrary  constants  of  the  integral 
equations  of  motion,  and  are  suffixed  to  the  differential  co- 

*  The  condition  that  the  forces  Flt  ...Fn  are  functions  of  q1 ,  . . .  qn  and 
alf  a2,  etc.,  which  last  are  functions  of  the  time,  is  analytically  equivalent 
to  the  condition  that  Flf  . . .  Fn  are  functions  of  qi,  ...qn  and  the  time. 
Explicit  mention  of  the  external  coordinates,  a1?  «2,  etc.,  has  been  made  in 
the  preceding  pages,  because  our  purpose  will  require  us  hereafter  to  con- 
sider these  coordinates  and  the  connected  forces,  Alt  A2,  etc.,  which  repre- 
sent the  action  of  the  systems  on  external  bodies. 


10  CONSERVATION  OF 

efficient  to  indicate  that  they  are  to  be  regarded  as  constant 
in  the  differentiation. 

We  may  give  to  this  principle  a  slightly  different  expres- 
sion.    Let  us  call  the  value  of  the  integral 


JT. 


.dpndqi...  dqn  (23) 


taken  within  any  limits  the  extension-in-phase  within  those 
limits. 

When  the  phases  bounding  an  extension-in-phase  vary  in 
the  course  of  time  according  to  the  dynamical  laws  of  a  system 
subject  to  forces  which  are  functions  of  the  coordinates  either 
alone  or  with  the  time,  the  value  of  the  extension-in-phase  thus 
bounded  remains  constant.  In  this  form  the  principle  may  be 
called  the  principle  of  conservation  of  extension-in-phase.  In 
some  respects  this  may  be  regarded  as  the  most  simple  state- 
ment of  the  principle,  since  it  contains  no  explicit  reference 
to  an  ensemble  of  systems. 

Since  any  extension-in-phase  may  be  divided  into  infinitesi- 
mal .portions,  it  is  only  necessary  to  prove  the  principle  for 
an  infinitely  small  extension.  The  number  of  systems  of  an 
ensemble  which  fall  within  the  extension  will  be  represented 
by  the  integral 


/  .  .  .  /  D  dp!  .  .  .  dp 


If  the  extension  is  infinitely  small,  we  may  regard  D  as  con- 
stant in  the  extension  and  write 

D  I  .  .  .  I  dpl  .  .  .  dpn  dq^  .  .  .  dqn 

for  the  number  of  systems.  The  value  of  this  expression  must 
be  constant  in  time,  since  no  systems  are  supposed  to  be 
created  or  destroyed,  and  none  can  pass  the  limits,  because 
the  motion  of  the  limits  is  identical  with  that  of  the  systems. 
But  we  have  seen  that  D  is  constant  in  time,  and  therefore 
the  integral 

I  .  .  .  /  fa  .  .  .  dpn  dql  .  .  .  dqn, 


EXTENSION-IN-PHASE.  11 

which  we  have  called  the  extension-in-phase,  is  also  constant 
in  time.* 

Since  the  system  of  coordinates  employed  in  the  foregoing 
discussion  is  entirely  arbitrary,  the  values  of  the  coordinates 
relating  to  any  configuration  and  its  immediate  vicinity  do 
not  impose  any  restriction  upon  the  values  relating  to  other 
configurations.  The  fact  that  the  quantity  which  we  have 
called  density-in-phase  is  constant  in  time  for  any  given  sys- 
tem, implies  therefore  that  its  value  is  independent  of  the 
coordinates  which  are  used  in  its  evaluation.  For  let  the 
density-in-phase  as  evaluated  for  the  same  time  and  phase  by 
one  system  of  coordinates  be  DI,  and  by  another  system  -Z>2'. 
A  system  which  at  that  time  has  that  phase  will  at  another 
time  have  another  phase.  Let  the  density  as  calculated  for 
this  second  time  and  phase  by  a  third  system  of  coordinates 
be  Zy.  Now  we  may  imagine  a  system  of  coordinates  which 
at  and  near  the  first  configuration  will  coincide  with  the  first 
system  of  coordinates,  and  at  and  near  the  second  configuration 
will  coincide  with  the  third  system  of  coordinates.  This  will 
give  Dj'  —  ^Y'-  Again  we  may  imagine  a  system  of  coordi- 
nates which  at  and  near  the  first  configuration  will  coincide 
with  the  second  system  of  coordinates,  and  at  and  near  the 

*  If  we  regard  a  phase  as  represented  by  a  point  in  space  of  2  n  dimen- 
sions, the  changes  which  take  place  in  the  course  of  time  in  our  ensemble  of 
systems  will  be  represented  by  a  current  in  such  space.  This  current  will 
be  steady  so  long  as  the  external  coordinates  are  not  varied.  In  any  case 
the  current  will  satisfy  a  law  which  in  its  various  expressions  is  analogous 
to  the  hydrodynamic  law  which  may  be  expressed  by  the  phrases  conserva- 
tion of  volumes  or  conservation  of  density  about  a  moving  point,  or  by  the  equation 


The  analogue  in  statistical  mechanics  of  this  equation,  viz., 


may  be  derived  directly  from  equations  (3)  or  (6),  and  may  suggest  such 
theorems  as  have  been  enunciated,  if  indeed  it  is  not  regarded  as  making 
them  intuitively  evident.  The  somewhat  lengthy  demonstrations  given 
above  will  at  least  serve  to  give  precision  to  the  notions  involved,  and 
familiarity  with  their  use. 


12  EXTENSION-IN-PHASE 

second  configuration  will  coincide  with  the  third  system  of 
coordinates.  This  will  give  D%  =  Ds".  We  have  therefore 
2V  =  2>J. 

It  follows,  or  it  may  be  proved  in  the  same  way,  that  the 
value  of  an  extension-in-phase  is  independent  of  the  system 
of  coordinates  which  is  used  in  its  evaluation.  This  may 
easily  be  verified  directly.  If  g1^  .  .  ,qn^  Qlt  .  .  .  Qn  are  two 
systems  of  coordinates,  and  Pi,  •  •  •  pn>  P\i  •  -  •  Pn  the  cor- 
responding momenta,  we  have  to  prove  that 

J'...Jdp1...dpndqi...dqn=j*...fdPl...dPndQ1...dQn,(2£) 

when  the  multiple  integrals  are  taken  within  limits  consisting 
of  the  same  phases.  And  this  will  be  evident  from  the  prin- 
ciple on  which  we  change  the  variables  in  a  multiple  integral, 
if  we  prove  that 

.  .  P.,  ft,  .  .  .  ft)  =  1 

>Pn>2i,  •-•  •  2V) 

where  the  first  member  of  the  equation  represents  a  Jacobian 
or  functional  determinant.  Since  all  its  elements  of  the  form 
dQ/dp  are  equal  to  zero,  the  determinant  reduces  to  a  product 
of  two,  and  we  have  to  prove  that 


d(Ql9 


We  may  transform  any  element  of  the  first  of  these  deter- 
minants as  follows.  By  equations  (2)  and  (3),  and  in 
view  of  the  fact  that  the  (j's  are  linear  functions  of  the  <?'s 
and  therefore  of  the  _p's,  with  coefficients  involving  the  <?'s, 
so  that  a  differential  coefficient  of  the  form  dQr/dpy  is  function 
of  the  <?'s  alone,  we  get  * 

*  The  form  of  the  equation 

d    dep  _    d    dfp 
dpy  dQx      dQx  dpv 

in  (27)  reminds  us  of  the  fundamental  identity  in  the  differential  calculus 
relating  to  the  order  of  differentiation  with  respect  to  independent  variables. 
But  it  will  be  observed  that  here  the  variables  Qx  and  py  are  not  independent 
and  that  the  proof  depends  on  the  linear  relation  between  the  Q's  and  the  p's. 


IS  AN  INVARIANT.  13 


r  dQx  dpy 

^^n/^dQL\=_d_de,==d^ 
dQx  r^i  W&  %J       d&  cZft,       d&  ' 


But  since  f'0 

r—  i  \  a  (j/r      / 

d-k  =  ^.  (28) 

*&     ^0. 

Therefore, 

...gn) 


...  Qn) 
The  equation  to  be  proved  is  thus  reduced  to 


which  is  easily  proved  by  the  ordinary  rule  for  the  multiplica- 
tion of  determinants. 

The  numerical  value  of  an  extension-in-phase  will  however 
depend  on  the  units  in  which  we  measure  energy  and  time. 
For  a  product  of  the  form  dp  dq  has  the  dimensions  of  energy 
multiplied  by  time,  as  appears  from  equation  (2),  by  which 
the  momenta  are  defined.  Hence  an  extension-in-phase  has 
the  dimensions  of  the  nth  power  of  the  product  of  energy 
and  time.  In  other  words,  it  has  the  dimensions  of  the  nth 
power  of  action,  as  the  term  is  used  in  the  '  principle  of  Least 
Action.' 

If  we  distinguish  by  accents  the  values  of  the  momenta 
and  coordinates  which  belong  to  a  time  ?,  the  unaccented 
letters  relating  to  the  time  £,  the  principle  of  the  conserva- 
tion of  extension-in-phase  may  be  written  v  *  <• 

//"»  /*  /% 

...  I  dpi  . . .  dpndqi . . .  dqn  =  I  ...  I  dpj  . . .  dpnfdqir . . ,  dqn'}  (31) 
*J  *J  *J 

or  more  briefly 

r  r       r 

>!7 . . .  dq^  (32) 


14  CONSERVATION  OF 

the  limiting  phases  being  those  which  belong  to  the  same 
systems  at  the  times  t  and  If  respectively.  But  we  have 
identically 


/.../*,...,  ,-/.. 


for  such  limits.    The  principle  of  conservation  of  extension-in- 
phase  may  therefore  be  expressed  in  the  form 

•    •    g«)  -,  xooN 

..g.9  =  1' 

This    equation   is    easily    proved   directly.      For    we    have 
identically 

d(Pl,...qn)  _    d(Pl,...qn) 


•  •  •  g.'O  <*(M  •  •  •  g.O  ' 

where  the  double  accents  distinguish  the  values  of  the  momenta 
and  coordinates  for  a  time  if'.  If  we  vary  t,  while  if  and  t" 
remain  constant,  we  have 

d_   d(Pl,  ...qn)    _  d(Pl"9  .  .  .  qn")  d_  d(Pl,  ...qn) 


Now  since  the  time  if'  is  entirely  arbitrary,  nothing  prevents 
us  from  making  if1  identical  with  t  at  the  moment  considered. 
Then  the  determinant 


•  •  -  ?»") 

will  have  unity  for  each  of  the  elements  on  the  principal 
diagonal,  and  zero  for  all  the  other  elements.  Since  every 
term  of  the  determinant  except  the  product  of  the  elements 
on  the  principal  diagonal  will  have  two  zero  factors,  the  differen- 
tial of  the  determinant  will  reduce  to  that  of  the  product  of 
these  elements,  i.  e.,  to  the  sum  of  the  differentials  of  these 
elements.  This  gives  the  equation 

d 


_. 
dt  d(pj>,  .  .  .  qn»)        dp,"  '  dpn"        dqj*  '  dqn» 

Now  since  t  =  t"  ,  the  double  accents  in  the  second  member 
of  this  equation  may  evidently  be  neglected.  This  will  give, 
in  virtue  of  such  relations  as  (16), 


EXTENSION-IN-PHASE.  15 

d    d(plt  ... 


dtd(Pl»,...yn") 

which  substituted  in  (34)  will  give 
d 


_ 
- 


...n    _ 

dtd(Pl',...qn') 

The  determinant  in  this  equation  is  therefore  a  constant,  the 
value  of  which  may  be  determined  at  the  instant  when  t  =  £', 
when  it  is  evidently  unity.  Equation  (33)  is  therefore 
demonstrated. 

Again,  if  we  write  a,  ...  h  for  a  system  of  2  n  arbitrary  con- 
stants of  the  integral  equations  of  motion,  pv  qv  etc.  will  be 
functions  of.  a,  ...  h,  and  t,  and  we  may  express  an  extension- 
in-phase  in  the  form 


/rd(p 
"V  «*(< 


,,       ^|T  da  -  -  •  dh-  (35> 

d(a,  ...h) 

If  we  suppose  the  limits  specified  by  values  of  a,  .  .  .  ^,  a 
system  initially  at  the  limits  will  remain  at  the  limits. 
The  principle  of  conservation  of  extension-in-phase  requires 
that  an  extension  thus  bounded  shall  have  a  constant  value. 
This  requires  that  the  determinant  under  the  integral  sign 
shall  be  constant,  which  may  be  written 


...n 
dt   d(a,...h)   =°*  (36) 

This  equation,  which  may  be  regarded  as  expressing  the  prin- 
ciple of  conservation  of  extension-in-phase,  may  be  derived 
directly  from  the  identity 

•  •  gj       <*(pi,  ...gn)  d(pi',  .  .  .  qnr) 


d(a,  ...h)    '     d(plf,  .  .  .  qn')     d(a,  ...  h) 
in  connection  with  equation  (33). 

Since  the  coordinates  and  momenta  are  functions  of  a,  ...  .  h, 
and  t,  the  determinant  in  (36)  must  be  a  function  of  the  same 
variables,  and  since  it  does  not  vary  with  the  time,  it  must 
be  a  function  of  a,  ...  h  alone.  We  have  therefore 

„...*).  '        (37) 


16  CONSERVATION  OF 

It  is  the  relative  numbers  of  systems  which  fall  within  dif- 
ferent limits,  rather  than  the  absolute  numbers,  with  which  we 
are  most  concerned.  It  is  indeed  only  with  regard  to  relative 
numbers  that  such  discussions  as  the  preceding  will  apply 
with  literal  precision,  since  the  nature  of  our  reasoning  implies 
that  the  number  of  systems  in  the  smallest  element  of  space 
which  we  consider  is  very  great.  This  is  evidently  inconsist- 
ent with  a  finite  value  of  the  total  number  of  systems,  or  of 
the  density-in-phase.  Now  if  the  value  of  D  is  infinite,  we 
cannot  speak  of  any  definite  number  of  systems  within  any 
finite  limits,  since  all  such  numbers  are  infinite.  But  the 
ratios  of  these  infinite  numbers  may  be  perfectly  definite.  If 
we  write  -ZVfor  the  total  number  of  systems,  and  set 

r  =  %.  (38) 

P  may  remain  finite,  when  JV*  and  D  become  infinite.     The 
integral 

"         *  ...  dqn  (39) 


taken  within  any  given  limits,  will  evidently  express  the  ratio 
of  the  number  of  systems  falling  within  those  limits  to  the 
whole  number  of  systems.  This  is  the  same  thing  as  the 
probability  that  an  unspecified  system  of  the  ensemble  (i.  e. 
one  of  which  we  only  know  that  it  belongs  to  the  ensemble) 
will  lie  within  the  given  limits.  The  product 

PdPl...dqn  (40) 

expresses  the  probability  that  an  unspecified  system  of  the 
ensemble  will  be  found  in  the  element  of  extension-in-phase 
dpi  .  .  .  dqn.  We  shall  call  P  the  coefficient  of  probability  of  the 
phase  considered.  Its  natural  logarithm  we  shall  call  the 
index  of  probability  of  the  phase,  and  denote  it  by  the  letter  77. 
If  we  substitute  NP  and  Ne1  for  D  in  equation  (19),  we  get 


and 


PROBABILITY  OF  PHASE.  17 

The  condition  of  statistical  equilibrium  may  be  expressed 
by  equating  to  zero  the  second  member  of  either  of  these 
equations. 

The  same  substitutions  in  (22)  give 

.,=°'  (43) 

(IX.... =°-  (44) 

That  is,  the  values  of  P  and  rj,  like  those  of  D,  are  constant 
in  time  for  moving  systems  of  the  ensemble.  From  this  point 
of  view,  the  principle  which  otherwise  regarded  has  been 
called  the  principle  of  conservation  of  density-in-phase  or 
conservation  of  extension-in-phase,  may  be  called  the  prin- 
ciple of  conservation  of  the  coefficient  (or  index)  of  proba- 
bility of  a  phase  varying  according  to  dynamical  laws,  or 
more  briefly,  the  principle  of  conservation  of  probability  of 
phase.  It  is  subject  to  the  limitation  that  the  forces  must  be 
functions  of  the  coordinates  of  the  system  either  alone  or  with 
the  time. 

The  application  of  this  principle  is  not  limited  to  cases  in 
which  there  is  a  formal  and  explicit  reference  to  an  ensemble  of 
systems.  Yet  the  conception  of  such  an  ensemble  may  serve 
to  give  precision  to  notions  of  probability.  It  is  in  fact  cus- 
tomary in  the  discussion  of  probabilities  to  describe  anything 
which  is  imperfectly  known  as  something  taken  at  random 
from  a  great  number  of  things  which  are  completely  described. 
But  if  we  prefer  to  avoid  any  reference  to  an  ensemble 
of  systems,  we  may  observe  that  the  probability  that  the 
phase  of  a  system  falls  within  certain  limits  at  a  certain  time, 
is  equal  to  the  probability  that  at  some  other  time  the  phase 
will  fall  within  the  limits  formed  by  phases  corresponding  to 
the  first.  For  either  occurrence  necessitates  the  other.  That 
is,  if  we  write  P'  for  the  coefficient  of  probability  of  the 
phase  pi,  •  •  •  qn'  at  the  time  ^,  and  P"  for  that  of  the  phase 
jp/',  .  .  .  qn"  at  the  time  tf', 

2 


18  CONSERVATION  OF 

J.  .  .  JV  dtf  .  .  .  dqj  =f.  .  .  Jp"  dp{'  .  .  .  dqn",      (45) 

where  the  limits  in  the  two  cases  are  formed  by  corresponding 
phases.  When  the  integrations  cover  infinitely  small  vari- 
ations of  the  momenta  and  coordinates,  we  may  regard  P*  and 
P"  as  constant  in  the  integrations  and  write 


P'f.  .  .fdPl>  •  •  •  <%»"  = 


Now  the  principle  of  the  conservation  of  extension-in-phase, 
which  has  been  proved  (viz.,  in  the  second  demonstration  given 
above)  independently  of  any  reference  to  an  ensemble  of 
systems,  requires  that  the  values  of  the  multiple  integrals  in 
this  equation  shall  be  equal.  This  gives 

P1'  =  Pf. 

With  reference  to  an  important  class  of  cases  this  principle 
may  be  enunciated  as  follows. 

When  the  differential  equations  of  motion  are  exactly  known, 
but  the  constants  of  the  integral  equations  imperfectly  deter- 
mined, the  coefficient  of  probability  of  any  phase  at  any  time  is 
equal  to  the  coefficient  of  probability  of  the  corresponding  phase 
at  any  other  time.  By  corresponding  phases  are  meant  those 
which  are  calculated  for  different  times  from  the  same  values 
of  the  arbitrary  constants  of  the  integral  equations. 

Since  the  sum  of  the  probabilities  of  all  possible  cases  is 
necessarily  unity,  it  is  evident  that  we  must  have 

all 

f...fpdPl...dqn  =  l,  (46) 

phases 

where  the  integration  extends  over  all  phases.  This  is  indeed 
only  a  different  form  of  the  equation 

811 


phases 

which  we  may  regard  as  defining 


PROBABILITY  OF  PHASE.  19 

The  values  of  the  coefficient  and  index  of  probability  of 
phase,  like  that  of  the  density-in-phase,  are  independent  of  the 
system  of  coordinates  which  is  employed  to  express  the  distri- 
bution in  phase  of  a  given  ensemble. 

In  dimensions,  the  coefficient  of  probability  is  the  reciprocal 
of  an  extension-in-phase,  that  is,  the  reciprocal  of  the  nth 
power  of  the  product  of  time  and  energy.  The  index  of  prob- 
ability is  therefore  affected  by  an  additive  constant  when  we 
change  our  units  of  time  and  energy.  If  the  unit  of  time  is 
multiplied  by  ct  and  the  unit  of  energy  is  multiplied  by  ce ,  all 
indices  of  probability  relating  to  systems  of  n  degrees  of 

freedom  will  be  increased  by  the  addition  of 

•"-- 
n  log  ct  +  n  log  c€.  (47) 


CHAPTER  II. 

APPLICATION  OF  THE  PRINCIPLE   OF  CONSERVATION 

OF  EXTENSION-IN-PHASE  TO  THE  THEORY 

OF  ERRORS. 

LET  us  now  proceed  to  combine  the  principle  which  has  been 
demonstrated  in  the  preceding  chapter  and  which  in  its  differ- 
ent applications  and  regarded  from  different  points  of  view 
has  been  variously  designated  as  the  conservation  of  density- 
in-phase,  or  of  extension-in-phase,  or  of  probability  of  phase, 
with  those  approximate  relations  which  are  generally  used  in 
the  'theory  of  errors.' 

We  suppose  that  the  differential  equations  of  the  motion  of 
a  system  are  exactly  known,  but  that  the  constants  of  the 
integral  equations  are  only  approximately  determined.  It  is 
evident  that  the  probability  that  the  momenta  and  coordinates 
at  the  time  t'  fall  between  the  limits  pj  and  pj  +  dp^  q^  and 
q-L  +  dq^  etc.,  may  be  expressed  by  the  formula 

e*  dPl' .  .  .  dqj,  (48) 

where  rf  (the  index  of  probability  for  the  phase  in  question)  is 
a  function  of  the  coordinates  and  momenta  and  of  the  time. 

Let  Qi,  P^t  etc.  be  the  values  of  the  coordinates  and  momenta 
which  give  the  maximum  value  to  ?/,  and  let  the  general 
value  of  rj  be  developed  by  Taylor's  theorem  according  to 
ascending  powers  and  products  of  the  differences  p^  —  P/, 
Q.I  ~  Ci'»  Qte">  an(i  let  us  suppose  that  we  have  a  sufficient 
approximation  without  going  beyond  terms  of  the  second 
degree  in  these  differences.  We  may  therefore  set 

n'  =  c  —  F',  (49) 

where  c  is  independent  of  the  differences  p^  —  P/,  q{  —  §/, 
etc.,  and  F1  is  a  homogeneous  quadratic  function  of  these 


THEORY  OF  ERRORS.  21 

differences.  The  terms  of  the  first  degree  vanish  in  virtue 
of  the  maximum  condition,  which  also  requires  that  F'  must 
have  a  positive  value  except  when  all  the  differences  men- 
tioned vanish.  If  we  set 

0=ef,  (50) 

we  may  write  for  the  probability  that  the  phase  lies  within 
the  limits  considered 


dPl>  .  .  .  dqj.  (51) 

C  is  evidently  the  maximum  value  of  the  coefficient  of  proba- 
bility at  the  time  considered. 

In  regard  to  the  degree  of  approximation  represented  by 
these  formulae,  it  is  to  be  observed  that  we  suppose,  as  is 
usual  in  the  'theory  of  errors/  that  the  determination  (ex- 
plicit or  implicit)  of  the  constants  of  motion  is  of  such 
precision  that  the  coefficient  of  probability  e*  or  Ce~F'  is 
practically  zero  except  for  very  small  values  of  the  differences 
Pi  —  P1/,  q^  —  Ci'>  e^c<  For  very  small  values  of  these 
differences  the  approximation  is  evidently  in  general  sufficient, 
for  larger  values  of  these  differences  the  value  of  Ce~F'  will 
be  sensibly  zero,  as  it  should  be,  and  in  this  sense  the  formula 
will  represent  the  facts. 

We  shall  suppose  that  the  forces  to  which  the  system  is 
subject  are  functions  of  the  coordinates  either  alone  or  with 
the  time.  The  principle  of  conservation  of  probability  of 
phase  will  therefore  apply,  which  requires  that  at  any  other 
time  (t")  the  maximum  value  of  the  coefficient  of  probability 
shall  be  the  same  as  at  the  time  t\  and  that  the  phase 
(Pi',  Qi'-)  etc.)  which  has  this  greatest  probability-coefficient, 
shall  be  that  which  corresponds  to  the  phase  (P/,  §-/,  etc.), 
i.  e.,  which  is  calculated  from  the  same  values  of  the  constants 
of  the  integral  equations  of  motion. 

We  may  therefore  write  for  the  probability  that  the  phase 
at  the  time  t"  falls  within  the  limits  p^1  and  p:"  +  dp^  #/' 
and  #/'  +  cfy/',  etc., 

"  dpi"  ...dqj',  (52) 


CONSERVATION  OF+EXTENSION-IN-PHASE 

where  C  represents  the  same  value  as  in  the  preceding 
formula,  viz.,  the  constant  value  of  the  maximum  coefficient 
of  probability,  and  Fn  is  a  quadratic  function  of  the  differences 
Pi  ~  pi">  <?i"  -  Ci",  etc.,  the  phase  (Px",  QJ'  etc.)  being  that 
which  at  the  time  t"  corresponds  to  the  phase  (P/,  #/,  etc.) 
at  the  tune  t'. 

Now  we  have  necessarily 


J*.  .  . 


&>i"  . . .  d£»"  =  1,  (53) 

when  the  integration  is  extended  over  all  possible  phases. 
It  will  be  allowable  to  set  ±  oo  for  the  limits  of  all  the  coor- 
dinates and  momenta,  not  because  these  values  represent  the 
actual  limits  of  possible  phases,  but  because  the  portions  of 
the  integrals  lying  outside  of  the  limits  of  all  possible  phases 
will  have  sensibly  the  value  zero.  With  ±  oo  for  limits,  the 
equation  gives 


l,  (64) 

Vf     Vf" 

where/'  is  the  discriminant  *  of  F1,  and/"  that  of  F".  This 
discriminant  is  therefore  constant  in  time,  and  like  C  an  abso- 
lute invariant  hi  respect  to  the  system  of  coordinates  which 
may  be  employed.  In  dimensions,  like  (72,  it  is  the  reciprocal 
of  the  2nth  power  of  the  product  of  energy  and  time. 

Let  us  see  precisely  how  the  functions  F'  and  F'f  are  related. 
The  principle  of  the  conservation  of  the  probability-coefficient 
requires  that  any  values  of  the  coordinates  and  momenta  at  the 
time  tf  shall  give  the  function  F'  the  same  value  as  the  corre- 
_  sponding  coordinates  and  momenta  at  the  time  tn  give  to  F". 
Therefore  Fn  may  be  derived  from  F'  by  substituting  for 
Pi*  •  •  -  9.n  their  values  in  terms  of  p^',  . . .  <?/'.  Now  we 
have  approximately 

*  This  term  is  used  to  denote  the  determinant  having  for  elements  on  the 
principal  diagonal  the  coefficients  of  the  squares  in  the  quadratic  function 
F',  and  for  its  other  elements  the  halves  of  the  coefficients  of  the  products 
inF'. 


AND   THEORY  OF  ERRORS.  23 


...+i^  (?."-<?."), 


(55) 


and  as  in  IF"  terms  of  higher  degree  than  the  second  are  to  be 
neglected,  these  equations  may  be  considered  accurate  for  the 
purpose  of  the  transformation  required.  Since  by  equation 
(33)  the  eliminant  of  these  equations  has  the  value  unity, 
the  discriminant  of  F"  will  be  equal  to  that  of  F1,  as  has 
already  appeared  from  the  consideration  of  the  principle  of 
conservation  of  probability  of  phase,  which  is,  in  fact,  essen- 
tially the  same  as  that  expressed  by  equation  (33). 
At  the  time  t\  the  phases  satisfying  the  equation 

F'  =  k,  (56) 

where  7c  is  any  positive  constant,  have  the  probability-coeffi- 
cient C  e~k .  At  the  time  tf",  the  corresponding  phases  satisfy 
the  equation 

F"  =  k9  (57) 

and  have  the  same  probability-coefficient.  So  also  the  phases 
within  the  limits  given  by  one  or  the  other  of  these  equations 
are  corresponding  phases,  and  have  probability-coefficients 
greater  than  C '  e~k,  while  phases  without  these  limits  have  less 
probability-coefficients.  The  probability  that  the  phase  at 
the  time  tf  falls  within  the  limits  F'  —  Jc  is  the  same  as  the 
probability  that  it  falls  within  the  limits  F"  =  k  at  the  time  t", 
since  either  event  necessitates  the  other.  This  probability 
may  be  evaluated  as  follows.  We  may  omit  the  accents,  as 
we  need  only  consider  a  single  time.  Let  us  denote  the  ex- 
tension-in-phase  within  the  limits  F  =  k  by  Z7,  and  the  prob- 
ability that  the  phase  falls  within  these  limits  by  R,  also  the 
extension-in-phase  within  the  limits  F  =  1  by  Ur  We  have 
then  by  definition 

F=k 

l...dqn,  (58) 


24          CONSERVATION  OF  EXTENSION-IN-PHASE 
F—k 


F=l 


But   since  F  is   a  homogeneous   quadratic   function   of   the 
differences 

we  have  identically 

F=k 

rt 

d(pi  -Pi)  .  .  .  d(qn  -  Qn) 
kF=k 

rwy&i 

F=l 

-Pj...d(<!.-Q1). 

That  is  U=knUl}  (61) 

whence  dU=  U1nkn~ldk.  (62) 

But  if  k  varies,  equations  (58)  and  (59)  give 

F=k-\-dk 

dU  =  I  .  .  .  I  dpi  . .  .  dqn  (63) 

F=k 

F=k+dk 

F=k 

Since  the  factor  Oe~F  has  the  constant  value   Ce~k  in  the 
last  multiple  integral,  we  have 

dR  =  C  e~kd  U  =  C  Ui  n  e~k  kn~l  dk,  (65) 

n  e-k  (\  +  &  +       +  .  .  .  +  N  +  const.       (66) 

We  may  determine  the  constant  of  integration  by  the  condition 
that  R  vanishes  with  k.     This  gives 


AND   THEORY  OF  ERRORS.  25 


(67) 


R  =  C  Z7i  ]n  -  C  U^  \n  e~k  fl  +  k  +  ~  +  .  .  .  +  r^jY 

We  may  determine  the  value  of  the  constant   U^  by  the  con- 
dition that  R  =  1  for  k  =  oo.     This  gives  (7  £7^  jw  ==  1,  and 

K  =  l  _  e-k(l  +  A;  +  ^  .  .  .  +  [^ZTfV  W 

^« 

(69) 


It  is  worthy  of  notice  that  the  form  of  these  equations  de- 
pends only  on  the  number  of  degrees  of  freedom  of  the  system, 
being  in  other  respects  independent  of  its  dynamical  nature, 
except  that  the  forces  must  be  functions  of  the  coordinates 
either  alone  or  with  the  time. 

If  we  write 

*»* 

for  the  value  of  k  which  substituted  in  equation  (68)  will  give 
R  =  1,  the  phases  determined  by  the  equation 

F--=kB=i  (70) 

will  have  the  following  properties. 

The  probability  that  the  phase  falls  within  the  limits  formed 
by  these  phases  is  greater  than  the  probability  that  it  falls 
within  any  other  limits  enclosing  an  equal  extension-in-phase. 
It  is  equal  to  the  probability  that  the  phase  falls  without  the 
same  limits. 

These  properties  are  analogous  to  those  which  in  the  theory 
of  errors  in  the  determination  of  a  single  quantity  belong  to 
values  expressed  by  A  ±  a,  when  A  is  the  most  probable 
value,  and  a  the  'probable  error.' 


CHAPTER  III. 

APPLICATION  OF  THE  PRINCIPLE  OF  CONSERVATION  OF 
EXTENSION-IN-PHASE  TO  THE  INTEGRATION  OF  THE 
DIFFERENTIAL  EQUATIONS  OF  MOTION.* 

WE  have  seen  that  the  principle  of  conservation  of  exten- 
sion-in-phase  may  be  expressed  as  a  differential  relation  be- 
tween the  coordinates  and  momenta  and  the  arbitrary  constants 
of  the  integral  equations  of  motion.  Now  the  integration  of 
the  differential  equations  of  motion  consists  in  the  determina- 
tion of  these  constants  as  functions  of  the  coordinates'  and 
momenta  with  the  time,  and  the  relation  afforded  by  the  prin- 
ciple of  conservation  of  extension-in-phase  may  assist  us  in 
this  determination. 

It  will  be  convenient  to  have  a  notation  which  shall  not  dis- 
tinguish between  the  coordinates  and  momenta.  If  we  write 
rx  .  .  .  r2n  for  the  coordinates  and  momenta,  and  a  ...  h  as  be- 
fore for  the  arbitrary  constants,  the  principle  of  which  we 
wish  to  avail  ourselves,  and  which  is  expressed  by  equation 
(37),  may  be  written 


,...*).  (71) 

Let  us  first  consider  the  case  in  which  the  forces  are  deter- 
mined by  the  coordinates  alone.  Whether  the  forces  are 
'  conservative  '  or  not  is  immaterial.  Since  the  differential 
equations  of  motion  do  not  contain  the  time  (t)  in  the  finite 
form,  if  we  eliminate  dt  from  these  equations,  we  obtain  2^  —  1 
equations  in  rl  ,  .  .  .  r2n  and  their  differentials,  the  integration 
of  which  will  introduce  2  n  —  1  arbitrary  constants  which  we 
shall  call  b  ...  h.  If  we  can  effect  these  integrations,  the 

*  See  Boltzmann:  "  Zusammenhang  zwischen  den  Satzen  iiber  das  Ver- 
halten  mehratomiger  Gasmoleciile  mit  Jacobi's  Princip  des  letzten  Multi- 
plicators.  Sitzb.  der  Wiener  Akad.,Bd.  LXIII,  Abth.  II.,  S.  679,  (1871). 


THEORY  OF  INTEGRATION.  27 

remaining  constant  (a)  will  then  be  introduced  in  the  final 
integration,  (viz.,  that  of  an  equation  containing  dt,}  and  will 
be  added  to  or  subtracted  from  t  in  the  integral  equation. 
Let  us  have  it  subtracted  from  t.  It  is  evident  then  that 


Moreover,  since  5,  ...  h  and  t  —  a  are  independent  functions 
of  rl  ,  .  .  .  r2n,  the  latter  variables  are  functions  of  the  former. 
The  Jacobian  in  (71)  is  therefore  function  of  6,  .  .  .  ^,  and 
t  —  a,  and  since  it  does  not  vary  with  t  it  cannot  vary  with  #. 
We  have  therefore  in  the  case  considered,  viz.,  where  the 
forces  are  functions  of  the  coordinates  alone, 


Now  let  us  suppose  that  of  the  first  2  n  —  1  integrations  we 
have  accomplished  all  but  one,  determining  2  n  —  2  arbitrary 
constants  (say  c?,  ...  h)  as  functions  of  r^  ,  .  .  .  r2n  ,  leaving  b  as 
well  as  a  to  be  determined.  Our  2  w  —  2  finite  equations  en- 
able us  to  regard  all  the  variables  r^  ,  .  .  .  r2n,  and  all  functions 
of  these  variables  as  functions  of  two  of  them,  (say  rl  and  r2,) 
with  the  arbitrary  constants  <?,...  h.  To  determine  5,  we 
have  the  following  equations  for  constant  values  of  <?,  ...  h. 


u-/  2  —  ~~; —  «**  T  ~77~  t*v* 

da  db 

df^i ,  r2)  c?7*2   7          dTi  .    . 

whence  -^7 — TT-  db  —  —  ^-  dr^-\-  -=—  «r2.  (74) 

d(a,  6)  c?a  c?a 

Now,  by  the  ordinary  formula  for  the  change  of  variables, 


=  r 

J 


zn) 


a^ 


28  CONSERVATION  OF  EXTENSION-IN-PHASE 

where  the  limits  of  the  multiple  integrals  are  formed  by  the 
same  phases.     Hence 

d(ri,rz)       d(r^  ...rZn)    d(c,  ...  h) 
d(a,b)   "    d(a,...h)    d(r99...rj 

With  the  aid  of  this  equation,  which  is  an  identity,  and  (72), 
we  may  write  equation  (74)  hi  the  form 


The  separation  of  the  variables  is  now  easy.  The  differen- 
tial equations  of  motion  give  rl  and  rz  in  terms  of  'r^  ,  .  .  .  r2n. 
The  integral  equations  already  obtained  give  <?,...  h  and 
therefore  the  Jacobian  d(c,  .  .  .  A)/c?(r3,  .  .  .  r2n),  in  terms  of 
the  same  variables.  But  in  virtue  of  these  same  integral 
equations,  we  may  regard  functions  of  r19  .  .  .  r2n  as  functions 
of  rl  and  r%  with  the  constants  c,  .  .  .  h.  If  therefore  we  write 
the  equation  in  the  form 

d(ri,  .  .  .r2n)       _  r2  ri  , 

'      ~  **-  ..h)  dr*>       (77) 


d(rs,  •  ..r2n)  d(r8,  .  .  .  r2n) 

the  coefficients  of  drl  and  dr%  may  be  regarded  as  known  func- 
tions of  rx  and  r2  with  the  constants  <?,...  h.  The  coefficient 
of  db  is  by  (73)  a  function  of  6,  .  .  .  h.  It  is  not  indeed  a 
known  function  of  these  quantities,  but  since  <?,...  h  are 
regarded  as  constant  in  the  equation,  we  know  that  the  first 
member  must  represent  the  differential  of  some  function  of 
5,  ...  A,  for  which  we  may  write  b'.  We  have  thus 

db'  =          r*          dr  ~  ..h)  dr*>  (78) 


d(r8,  .  ..ran)  d(r8,  ...r2n) 

which  may  be  integrated  by  quadratures  and  gives  V  as  func- 
tions of  r1?  r2 ,  ...<?,...  A,  and  thus  as  function  of  r1?  .  .  .  r2n. 
This  integration  gives  us  the  last  of  the  arbitrary  constants 
which  are  functions  of  the  coordinates  and  momenta  without 
the  time.  The  final  integration,  which  introduces  the  remain- 


AND   THEORY  OF  INTEGRATION.  29 

ing  constant  (a),  is  also  a  quadrature,  since  the  equation  to 
be  integrated  may  be  expressed  in  the  form 


Now,  apart  from  any  §uch  considerations  as  have  been  ad- 
duced, if  we  limit  ourselves  to  the  changes  which  take  place 
in  time,  we  have  identically 

r2  dr±  —  r^  drz  =  0, 

and  r±  and  r2  are  given  in  terms  of  rv  .  .  .  r2n  by  the  differential 
equations  of  motion.  When  we  have  obtained  2  n  —  2  integral 
equations,  we  may  regard  r2  and  r^  as  known  functions  of  rl 
and  r2  .  The  only  remaining  difficulty  is  in  integrating  this 
equation.  If  the  case  is  so  simple  as  to  present  no  difficulty, 
or  if  we  have  the  skill  or  the  good  fortune  to  perceive  that  the 
multiplier 


d(c,...h)   '  (79) 

d(r.,...rfc) 

or  any  other,  will  make  the  first  member  of  the  equation  an 
exact  differential,  we  have  no  need  of  the  rather  lengthy  con- 
siderations which  have  been  adduced.  The  utility  of  the 
principle  of  conservation  of  extension-in-phase  is  that  it  sup- 
plies a  '  multiplier '  which  renders  the  equation  integrable,  and 
which  it  might  be  difficult  or  impossible  to  find  otherwise. 

It  will  be  observed  that  the  function  represented  by  b'  is  a 
particular  case  of  that  represented  by  b.  The  system  of  arbi- 
trary constants  «,  5',  c . . .  h  has  certain  properties  notable  for 
simplicity.  If  we  write  b'  for  b  in  (77),  and  compare  the 
result  with  (78),  we  get 

=  1.  (80) 


d(a,  b',  c,  .  .  .  A) 
Therefore  the  multiple  integral 


da  dbf  do  .  .  .  dh  (81) 


30          CONSERVATION  OF  EXTENSION-IN-PHASE 

taken  within  limits  formed  by  phases  regarded  as  contempo- 
raneous represents  the  extension-in-phase  within  those  limits. 

The  case  is  somewhat  different  when  the  forces  are  not  de- 
termined by  the  coordinates  alone,  but  are  functions  of  the 
coordinates  with  the  time.  All  the  arbitrary  constants  of  the 
integral  equations  must  then  be  regarded  in  the  general  case 
as  functions  of  rv  . . .  r2n,  and  t.  We  cannot  use  the  princi- 
ple of  conservation  of  extension-in-phase  until  we  have  made 
2n  —  ~L  integrations.  Let  us  suppose  that  the  constants  6,  ...  h 
have  been  determined  by  integration  in  terms  of  rv  . . .  r2w,  and 
t,  leaving  a  single  constant  (a)  to  be  thus  determined.  Our 
2  %  —  1  finite  equations  enable  us  to  regard  all  the  variables 
rv  . . .  r2n  as  functions  of  a  single  one,  say  rr 

For  constant  values  of  5, ...  A,  we  have 

**-£*  + ft*  (82) 

Now 

*  *       \MI  1          ,  _ 

-5—  da  dr*  .  .  .  drzn  = 

t 

da  .    .  dh 


d(a,  ...h) 

^"^    I     •   •    •     f    " 

J        J    d(a}  ...  A)    d(r2,  .  .  .  r2n) 

where  the   limits  of  the  integrals  are   formed  by  the  same 
phases.     We  have  therefore 

^'•••A>,  (83) 


da  "   d(a,...h)  d(rt,  .  .  .  r,n) 
by  which  equation  (82)  may  be  reduced  to  the  form 


da  = 


M         M 
a,  .  .  .  h)  d(b,  ...  A) 


d(r2,  .  .  . 


Now  we  know  by  (71)  that  the  coefficient  of  da  is  a  func- 
tion of  a,  ...  h.  Therefore,  as  £,  ...  h  are  regarded  as  constant 
in  the  equation,  the  first  number  represents  the  differential 


AND   THEORY  OF  INTEGRATION.  31 

of  a  function  of  a,  . . .  h,  which  we  may  denote  by  a'.  We 
have  then 

da'=   d(b,...h)   dr^~    d(b*..K)    dt>  (85) 

dfa,  ...r2n)  d(r2,  ...r2n) 

which  may  be  integrated  by  quadratures.  In  this  case  we 
may  say  that  the  principle  of  conservation  of  extension-in- 
phase  has  supplied  the  *  multiplier ' 

1 

d(b,  ...h)  (86) 

d(rz,  . . .  rzn) 

for  the  integration  of  the  equation 

dr,  -rldt  =  0.  (87) 

The  system  of  arbitrary  constants  a',  5, ...  h  has  evidently 
the  same  properties  which  were  noticed  in  regard  to  the 
system  a,  6', ...  h. 


CHAPTER  IV. 

ON  THE  DISTRIBUTION  IN  PHASE  CALLED  CANONICAL, 
IN  WHICH  THE  INDEX  OF  PROBABILITY  IS  A  LINEAR 
FUNCTION  OF  THE  ENERGY. 

LET  us  now  give  our  attention  to  the  statistical  equilibrium 
of  ensembles  of  conservation  systems,  especially  to  those  cases 
and  properties  which  promise  to  throw  light  on  the  phenom- 
ena of  thermodynamics. 

The  condition  of  statistical  equilibrium  may  be  expressed 
in  the  form* 


where  P  is  the  coefficient  of  probability,  or  the  quotient  of 
the  density-in-phase  by  the  whole  number  of  systems.  To 
satisfy  this  condition,  it  is  necessary  and  sufficient  that  P 
should  be  a  function  of  the  p's  and  q*s  (the  momenta  and 
coordinates)  which  does  not  vary  with  the  time  in  a  moving 
system.  In  all  cases  which  we  are  now  considering,  the 
energy,  or  any  function  of  the  energy,  is  such  a  function. 

P  =  f  unc.  (e) 

will  therefore  satisfy  the  equation,  as  indeed  appears  identi- 
cally if  we  write  it  in  the  form 


<Wd^_dP_de\  =0 
dq1dpl       dpldql)~ 


There  are,  however,  other  conditions  to  which  P  is  subject, 
which  are  not  so  much  conditions  of  statistical  equilibrium,  as 
conditions  implicitly  involved  in  the  definition  of  the  coeffi- 

*  See  equations  (20),  (41),  (42),  also  the  paragraph  following  equation  (20). 
The  positions  of  any  external  bodies  which  can  affect  the  systems  are  here 
supposed  uniform  for  all  the  systems  and  constant  in  time. 


J. 


CANONICAL  DISTRIBUTION.  33 

cient  of  probability,  whether  the  case  is  one  of  equilibrium 
or  not.  These  are:  that  P  should  be  single-valued,  and 
neither  negative  nor  imaginary  for  any  phase,  and  that  ex- 
pressed by  equation  (46),  viz., 

all 

JP4>...-  <*?»  =  !.  (89) 

phases 

These  considerations  exclude 

P  =  e  X  constant, 

as  well  as 

P  =  constant, 

as  cases  to  be  considered. 

The  distribution  represented  by 

(90) 


or 


where  ®  and  i/r  are  constants,  and  %  positive,  seems  to  repre- 
sent the  most  simple  case  conceivable,  since  it  has  the  property 
that  when  the  system  consists  of  parts  with  separate  energies, 
the  laws  of  the  distribution  in  phase  of  the  separate  parts  are 
of  the  same  nature,  —  a  property  which  enormously  simplifies 
the  discussion,  and  is  the  foundation  of  extremely  important 
relations  to  thermodynamics.  The  case  is  not  rendered  less 
simple  by  the  divisor  ®,  (a  quantity  of  the  same  dimensions  as 
e,)  but  the  reverse,  since  it  makes  the  distribution  independent 
of  the  units  employed.  The  negative  sign  of  e  is  required  by 
(89),  which  determines  also  the  value  of  ^  for  any  given 
©,  viz., 

all  f 

~® 


=f.  .  .f 


e      dp,...  dqn  .  (92) 

phases 

When  an  ensemble  of  systems  is  distributed  in  phase  in  the 
manner  described,  i.  e.^  when  the  index  of  probability  is   a 

3 


34  CANONICAL  DISTRIBUTION 

linear  function  of  the  energy,  we  shall  say  that  the  ensemble  is 
canonically  distributed,  and  shall  call  the  divisor  of  the  energy 
(®)  the  modulus  of  distribution. 

The  fractional  part  of  an  ensemble  canonically  distributed 
which  lies  within  any  given  limits  of  phase  is  therefore  repre- 
sented by  the  multiple  integral 


9  dpl  .  .  .  dqn  (93) 

taken  within  those  limits.  We  may  express  the  same  thing 
by  saying  that  the  multiple  integral  expresses  the  probability 
that  an  unspecified  system  of  the  ensemble  (i.  e.,  one  of 
which  we  only  know  that  it  belongs  to  the  ensemble)  falls 
within  the  given  limits. 

Since  the  value  of  a  multiple  integral  of  the  form  (23) 
(which  we  have  called  an  extension-in-phase)  bounded  by  any 
given  phases  is  independent  of  the  system  of  coordinates  by 
which  it  is  evaluated,  the  same  must  be  true  of  the  multiple 
integral  in  (92),  as  appears  at  once  if  we  divide  up  this 
integral  into  parts  so  small  that  the  exponential  factor  may  be 
regarded  as  constant  in  each.  The  value  of  ^r  is  therefore  in- 
dependent of  the  system  of  coordinates  employed. 

It  is  evident  that  ty  might  be  defined  as  the  energy  for 
which  the  coefficient  of  probability  of  phase  has  the  value 
unity.  Since  however  this  coefficient  has  the  dimensions  of 
the  inverse  nth  power  of  the  product  of  energy  and  time,* 
the  energy  represented  by  -\Jr  is  not  independent  of  the  units 
of  energy  and  time.  But  when  these  units  have  been  chosen, 
the  definition  of  ^  will  involve  the  same  arbitrary  constant  as 
e,  so  that,  while  in  any  given  case  the  numerical  values  of 
^r  or  e  will  be  entirely  indefinite  until  the  zero  of  energy  has 
also  been  fixed  for  the  system  considered,  the  difference  ty  —  e 
will  represent  a  perfectly  definite  amount  of  energy,  which  is 
entirely  independent  of  the  zero  of  energy  which  we  may 
choose  to  adopt. 

*  See  Chapter  I,  p.  19. 


OF  AN  ENSEMBLE  OF  SYSTEMS.  35 

It  is  evident  that  the  canonical  distribution  is  entirely  deter- 
mined by  the  modulus  (considered  as  a  quantity  of  energy) 
and  the  nature  of  the  system  considered,  since  when  equation 
(92)  is  satisfied  the  value  of  the  multiple  integral  (93)  is 
independent  of  the  units  and  of  the  coordinates  employed,  and 
of  the  zero  chosen  for  the  energy  of  the  system. 

In  treating  of  the  canonical  distribution,  we  shall  always 
suppose  the  multiple  integral  in  equation  (92)  to  have  a 
finite  value,  as  otherwise  the  coefficient  of  probability  van- 
ishes, and  the  law  of  distribution  becomes  illusory.  This  will 
exclude  certain  cases,  but  not  such  apparently,  as  will  affect 
the  value  of  our  results  with  respect  to  their  bearing  on  ther- 
modynamics. It  will  exclude,  for  instance,  cases  in  which  the 
system  or  parts  of  it  can  be  distributed  in  unlimited  space 
(or  in  a  space  which  has  limits,  but  is  still  infinite  in  volume), 
while  the  energy  remains  beneath  a  finite  limit.  It  also 
excludes  many  cases  in  which  the  energy  can  decrease  without 
limit,  as  when  the  system  contains  material  points  which 
attract  one  another  inversely  as  the  squares  of  their  distances. 
Cases  of  material  points  attracting  each  other  inversely  as  the 
distances  would  be  excluded  for  some  values  of  ®,  and  not 
for  others.  The  investigation  of  such  points  is  best  left  to 
the  particular  cases.  For  the  purposes  of  a  general  discussion, 
it  is  sufficient  to  call  attention  to  the  assumption  implicitly 
involved  in  the  formula  (92).* 

The  modulus  ©  has  properties  analogous  to  those  of  tem- 
perature in  thermodynamics.  Let  the  system  A  be  defined  as 
one  of  an  ensemble  of  systems  of  m  degrees  of  freedom 
distributed  in  phase  with  a  probability-coefficient 

*£% 

e    0    , 

*  It  will  be  observed  that  similar  limitations  exist  in  thermodynamics.  In 
order  that  a  mass  of  gas  can  be  in  thermodynamic  equilibrium,  it  is  necessary 
that  it  be  enclosed.  There  is  no  thermodynamic  equilibrium  of  a  (finite)  mass 
of  gas  in  an  infinite  space.  Again,  that  two  attracting  particles  should  be 
able  to  do  an  infinite  amount  of  work  in  passing  from  one  configuration 
(which  is  regarded  as  possible)  to  another,  is  a  notion  which,  although  per- 
fectly intelligible  in  a  mathematical  formula,  is  quite  foreign  to  our  ordinary 
conceptions  of  matter. 


36  CANONICAL  DISTRIBUTION 

and  the  system  B  as  one  of  an  ensemble  of  systems  of  n 
degrees  of  freedom  distributed  in  phase  with  a  probability- 
coefficient 


which  has  the  same  modulus.  Let  qv  .  .  .qm,  pv  .  .  .  pm  be  the 
coordinates  and  momenta  of  A,  and  qm+l  ,  .  .  .  qm+n,  pm+l  ,  .  .  .  pm+n 
those  of  £.  Now  we  may  regard  the  systems  A  and  B  as 
together  forming  a  system  0,  having  m  +  n  degrees  of  free- 
dom, and  the  coordinates  and  momenta  q^  .  .  .  <?,„+„,  pv  .  .  .  pm+n. 
The  probability  that  the  phase  of  the  system  (7,  as  thus  defined, 
will  fall  within  the  limits 

dpi  ,  .  .  .  dpm+n,  dq1  ,  .  .  .  dqm+n 

is  evidently  the  product  of  the  probabilities  that  the  systems 
A  and  B  will  each  fall  within  the  specified  limits,  viz., 


(94) 


We  may  therefore  regard  C  as  an  undetermined  system  of  an 
ensemble  distributed  with  the  probability-coefficient 


(95) 


an  ensemble  which  might  be  defined  as  formed  by  combining 
each  system  of  the  first  ensemble  with  each  of  the  second. 
But  since  eA  +  €B  is  the  energy  of  the  whole  system,  and 
^A  and  >/r  B  are  constants,  the  probability-coefficient  is  of  the 
general  form  which  we  are  considering,  and  the  ensemble  to 
which  it  relates  is  in  statistical  equilibrium  and  is  canonically 
distributed. 

This  result,  however,  so  far  as  statistical  equilibrium  is 
concerned,  is  rather  nugatory,  since  conceiving  of  separate 
systems  as  forming  a  single  system  does  not  create  any  in- 
teraction between  them,  and  if  the  systems  combined  belong  to 
ensembles  in  statistical  equilibrium,  to  say  that  the  ensemble 
formed  by  such  combinations  as  we  have  supposed  is  in  statis- 
tical equilibrium,  is  only  to  repeat  the  data  in  different 


OF  AN  ENSEMBLE  OF  SYSTEMS.         37 

words.  Let  us  therefore  suppose  that  in  forming  the  system 
C  we  add  certain  forces  acting  between  A  and  .5,  and  having 
the  force-function  —  eAB.  The  energy  of  the  system  C  is  now 
€A  +  €B  +  €ABI  and  an  ensemble  of  such  systems  distributed 
with  a  density  proportional  to 


(96) 


would  be  in  statistical  equilibrium.  Comparing  this  with  the 
probability-coefficient  of  C  given  above  (95),  we  see  that  if 
we  suppose  eAB  (or  rather  the  variable  part  of  this  term  when 
we  consider  all  possible  configurations  of  the  systems  A  and  B) 
to  be  infinitely  small,  the  actual  distribution  in  phase  of  C 
will  differ  infinitely  little  from  one  of  statistical  equilibrium, 
which  is  equivalent  to  saying  that  its  distribution  in  phase 
will  vary  infinitely  little  even  in  a  time  indefinitely  prolonged.* 
The  case  would  be  entirely  different  if  A  and  B  belonged  to 
ensembles  having  different  moduli,  say  ®A  and  ®5.  The  prob- 
ability-coefficient of  C  would  then  be 


which  is  not  approximately  proportional  to  any  expression  of 
the  form  (96). 

Before  proceeding  farther  in  the  investigation  of  the  dis- 
tribution in  phase  which  we  have  called  canonical,  it  will  be 
interesting  to  see  whether  the  properties  with  respect  to 

*  It  will  be  observed  that  the  above  condition  relating  to  the  forces  which 
act  between  the  different  systems  is  entirely  analogous  to  that  which  must 
hold  in  the  corresponding  case  in  thermodynamics.  The  most  simple  test 
of  the  equality  of  temperature  of  two  bodies  is  that  they  remain  in  equilib- 
rium when  brought  into  thermal  contact.  Direct  thermal  contact  implies 
molecular  forces  acting  between  the  bodies.  Now  the  test  will  fail  unless 
the  energy  of  these  forces  can  be  neglected  in  comparison  with  the  other 
energies  of  the  bodies.  Thus,  in  the  case  of  energetic  chemical  action  be- 
tween the  bodies,  or  when  the  number  of  particles  affected  by  the  forces 
acting  between  the  bodies  is  not  negligible  in  comparison  with  the  whole 
number  of  particles  (as  when  the  bodies  have  the  form  of  exceedingly  thin 
sheets),  the  contact  of  bodies  of  the  same  temperature  may  produce  con- 
siderable thermal  disturbance,  and  thus  fail  to  afford  a  reliable  criterion  of 
the  equality  of  temperature. 


38  OTHER  DISTRIBUTIONS 

statistical  equilibrium  which  have  been  described  are  peculiar 
to  it,  or  whether  other  distributions  may  have  analogous 
properties. 

Let  rjr  and  77"  be  the  indices  of  probability  in  two  independ- 
ent ensembles  which  are  each  in  statistical  equilibrium,  then 
rf  _j_  y  wni  De  the  index  in  the  ensemble  obtained  by  combin- 
ing each  system  of  the  first  ensemble  with  each  system  of  the 
second.  This  third  ensemble  will  of  course  be  in  statistical 
equilibrium,  and  the  function  of  phase  vf  +  if1  will  be  a  con- 
stant of  motion.  Now  when  infinitesimal  forces  are  added  to 
the  compound  systems,  if  r/  +  rf1  or  a  function  differing 
infinitesimally  from  this  is  still  a  constant  of  motion,  it  must 
be  on  account  of  the  nature  of  the  forces  added,  or  if  their  action 
is  not  entirely  specified,  on  account  of  conditions  to  which 
they  are  subject.  Thus,  in  the  case  already  considered, 
V  +  ??"  is  a  function  of  the  energy  of  the  compound  system, 
and  the  infinitesimal  forces  added  are  subject  to  the  law  of 
conservation  of  energy. 

Another  natural  supposition  in  regard  to  the  added  forces 
is  that  they  should  be  such  as  not  to  affect  the  moments  of 
momentum  of  the  compound  system.  To  get  a  case  in  which 
moments  of  momentum  of  the  compound  system  shall  be 
constants  of  motion,  we  may  imagine  material  particles  con- 
tained in  two  concentric  spherical  shells,  being  prevented  from 
passing  the  surfaces  bounding  the  shells  by  repulsions  acting 
always  in  lines  passing  through  the  common  centre  of  the 
shells.  Then,  if  there  are  no  forces  acting  between  particles  in 
different  shells,  the  mass  of  particles  in  each  shell  will  have, 
besides  its  energy,  the  moments  of  momentum  about  three 
axes  through  the  centre  as  constants  of  motion. 

Now  let  us  imagine  an  ensemble  formed  by  distributing  in 
phase  the  system  of  particles  in  one  shell  according  to  the 
index  of  probability 

•     ^-I+|+S+S'  (98) 

where  e  denotes  the  energy  of  the  system,  and  ©j ,  o>2 ,  &>3 ,  its 
three  moments  of  momentum,  and  the  other  letters  constants. 


HAVE  ANALOGOUS  PROPERTIES.  39 

In  like  manner  let  us  imagine  a  second  ensemble  formed  by 
distributing  in  phase  the  system  of  particles  in  the  other  shell 
according  to  the  index 


where  the  letters  have  similar  significations,  and  O,  Ox  ,  O2  ,  113 
the  same  values  as  in  the  preceding  formula.  Each  of  the 
two  ensembles  will  evidently  be  in  statistical  equilibrium,  and 
therefore  also  the  ensemble  of  compound  systems  obtained  by 
combining  each  system  of  the  first  ensemble  with  each  of  the 
second.  In  this  third  ensemble  the  index  of  probability  will  be 

k  +  ^-!±^  +  SL±^  +  2d^  +  a±3L-,   (ioo) 

vy  i/j  1/2  »*a 

where  the  four  numerators  represent  functions  of  phase  which 
are  constants  of  motion  for  the  compound  systems. 

Now  if  we  add  in  each  system  of  this  third  ensemble  infini- 
tesimal conservative  forces  of  attraction  or  repulsion  between 
particles  in  different  shells,  determined  by  the  same  law  for 
all  the  systems,  the  functions  o^  +  &>',  &>2  +  o>2',  and  &>3  +  w3' 
will  remain  constants  of  motion,  and  a  function  differing  in- 
finitely little  from  el  +  e  will  be  a  constant  of  motion.  It 
would  therefore  require  only  an  infinitesimal  change  in  the 
distribution  in  phase  of  the  ensemble  of  compound  systems  to 
make  it  a  case  of  statistical  equilibrium.  These  properties  are 
entirely  analogous  to  those  of  canonical  ensembles.* 

Again,  if  the  relations  between  the  forces  and  the  coordinates 
can  be  expressed  by  linear  equations,  there  will  be  certain 
"  normal  "  types  of  vibration  of  which  the  actual  motion  may 
be  regarded  as  composed,  and  the  whole  energy  may  be  divided 

*  It  would  not  be  possible  to  omit  the  term  relating  to  energy  in  the  above 
indices,  since  without  this  term  the  condition  expressed  by  equation  (89) 
cannot  be  satisfied. 

The  consideration  of  the  above  case  of  statistical  equilibrium  may  be 
made  the  foundation  of  the  theory  of  the  thermodynamic  equilibrium  of 
rotating  bodies,  —  a  subject  which  has  been  treated  by  Maxwell  in  his  memoir 
"  On  Boltzmann's  theorem  on  the  average  distribution  of  energy  in  a  system 
of  material  points."  Cambr.  Phil.  Trans.,  vol.  XII,  p.  547,  (1878). 


40  OTHER  DISTRIBUTIONS 

into  parts  relating  separately  to  vibrations  of  these  different 
types.  These  partial  energies  will  be  constants  of  motion, 
and  if  such  a  system  is  distributed  according  to  an  index 
which  is  any  function  of  the  partial  energies,  the  ensemble  will 
be  in  statistical  equilibrium.  Let  the  index  be  a  linear  func- 
tion of  the  partial  energies,  say 


Let  us  suppose  that  we  have  also  a  second  ensemble  com- 
posed of  systems  in  which  the  forces  are  linear  functions  of 
the  coordinates,  and  distributed  in  phase  according  to  an  index 
which  is  a  linear  function  of  the  partial  energies  relating  to 
the  normal  types  of  vibration,  say 

^~i?'*'~if  (102) 

Since  the  two  ensembles  are  both  in  statistical  equilibrium, 
the  ensemble  formed  by  combining  each  system  of  the  first 
with  each  system  of  the  second  will  also  be  in  statistical 
equilibrium.  Its  distribution  in  phase  will  be  represented  by 
the  index 


and  the  partial  energies  represented  by  the  numerators  in  the 
formula  will  be  constants  of  motion  of  the  compound  systems 
which  form  this  third  ensemble. 

Now  if  we  add  to  these  compound  systems  infinitesimal 
forces  acting  between  the  component  systems  and  subject  to 
the  same  general  law  as  those  already  existing,  viz.,  that  they 
are  conservative  and  linear  functions  of  the  coordinates,  there 
will  still  be  n  +  m  types  of  normal  vibration,  and  n  +  m 
partial  energies  which  are  independent  constants  of  motion. 
If  all  the  original  n  +  m  normal  types  of  vibration  have  differ- 
ent periods,  the  new  types  of  normal  vibration  will  differ  infini- 
tesimally  from  the  old,  and  the  new  partial  energies,  which  are 
constants  of  motion,  will  be  nearly  the  same  functions  of 
phase  as  the  old.  Therefore  the  distribution  in  phase  of  the 


HAVE  ANALOGOUS  PROPERTIES.  41 

ensemble  of  compound  systems  after  the  addition  of  the  sup- 
posed infinitesimal  forces  will  differ  infinitesimally  from  one 
which  would  be  in  statistical  equilibrium. 

The  case  is  not  so  simple  when  some  of  the  normal  types  of 
motion  have  the  same  periods.  In  this  case  the  addition  of 
infinitesimal  forces  may  completely  change  the  normal  types 
of  motion.  But  the  sum  of  the  partial  energies  for  all  the 
original  types  of  vibration  which  have  any  same  period,  will 
be  nearly  identical  (as  a  function  of  phase,  i.  e.,  of  the  coordi- 
nates and  momenta,)  with  the  sum  of  the  partial  energies  for 
the  normal  types  of  vibration  which  have  the  same,  or  nearly 
the  same,  period  after  the  addition  of  the  new  forces.  If, 
therefore,  the  partial  energies  in  the  indices  of  the  first  two 
ensembles  (101)  and  (102)  which  relate  to  types  of  vibration 
having  the  same  periods,  have  the  same  divisors,  the  same  will 
be  true  of  the  index  (103)  of  the  ensemble  of  compound  sys- 
tems, and  the  distribution  represented  will  differ  infinitesimally 
from  one  which  would  be  in  statistical  equilibrium  after  the 
addition  of  the  new  forces.* 

The  same  would  be  true  if  in  the  indices  of  each  of  the 
original  ensembles  we  should  substitute  for  the  term  or  terms 
relating  to  any  period  which  does  not  occur  in  the  other  en- 
semble, any  function  of  the  total  energy  related  to  that  period, 
subject  only  to  the  general  limitation  expressed  by  equation 
(89).  But  in  order  that  the  ensemble  of  compound  systems 
(with  the  added  forces)  shall  always  be  approximately  in 
statistical  equilibrium,  it  is  necessary  that  the  indices  of  the 
original  ensembles  should  be  linear  functions  of  those  partial 
energies  which  relate  to  vibrations  of  periods  common  to  the 
two  ensembles,  and  that  the  coefficients  of  such  partial  ener- 
gies should  be  the  same  in  the  two  indices.f 

*  It  is  interesting  to  compare  the  above  relations  with  the  laws  respecting 
the  exchange  of  energy  between  bodies  by  radiation,  although  the  phenomena 
of  radiations  lie  entirely  without  the  scope  of  the  present  treatise,  in  which 
the  discussion  is  limited  to  systems  of  a  finite  number  of  degrees  of  freedom. 

t  The  above  may  perhaps  be  sufficiently  illustrated  by  the  simple  case 
where  n  =  1  in  each  system.  If  the  periods  are  different  in  the  two  systems, 
they  may  be  distributed  according  to  any  functions  of  the  energies :  but  if 


42  CANONICAL  DISTRIBUTION 

The  properties  of  canonically  distributed  ensembles  of 
systems  with  respect  to  the  equilibrium  of  the  new  ensembles 
which  may  be  formed  by  combining  each  system  of  one  en- 
semble with  each  system  of  another,  are  therefore  not  peculiar 
to  them  in  the  sense  that  analogous  properties  do  not  belong 
to  some  other  distributions  under  special  limitations  in  regard 
to  the  systems  and  forces  considered.  Yet  the  canonical 
distribution  evidently  constitutes  the  most  simple  case  of  the 
kind,  and  that  for  which  the  relations  described  hold  with  the 
least  restrictions. 

Returning  to  the  case  of  the  canonical  distribution,  we 
shall  find  other  analogies  with  thermodynamic  systems,  if  we 
suppose,  as  in  the  preceding  chapters,*  that  the  potential 
energy  (eq)  depends  not  only  upon  the  coordinates  ql  .  .  .  qn 
which  determine  the  configuration  of  the  system,  but  also 
upon  certain  coordinates  «i,  «2,  etc.  of  bodies  which  we  call 
external?  meaning  by  this  simply  that  they  are  not  to  be  re- 
garded as  forming  any  part  of  the  system,  although  their 
positions  affect  the  forces  which  act  on  the  system.  The 
forces  exerted  by  the  system  upon  these  external  bodies  will 
be  represented  by  —  deqjdav  —  deqfda2,  etc.,  while  —  deqjdqv 
...  —  deq/dqn  represent  all  the  forces  acting  upon  the  bodies 
of  the  system,  including  those  which  depend  upon  the  position 
of  the  external  bodies,  as  well  as  those  which  depend  only 
upon  the  configuration  of  the  system  itself.  It  will  be  under- 
stood that  €p  depends  only  upon  qi  ,  .  .  .  qn  ,  p\  ,  .  .  .  pn  ,  in  other 
words,  that  the  kinetic  energy  of  the  bodies  which  we  call 
external  forms  no  part  of  the  kinetic  energy  of  the  system. 
It  follows  that  we  may  write 


although  a  similar  equation  would  not  hold  for  differentiations 
relative  to  the  internal  coordinates. 

the  periods  are  the  same  they  must  be  distributed  canonically  with  same 
modulus  in  order  that  the  compound  ensemble  with  additional  forces  may 
be  in  statistical  equilibrium. 
*  See  especially  Chapter  I,  p.  4. 


OF  AN  ENSEMBLE  OF  SYSTEMS.  43 

We  always  suppose  these  external  coordinates  to  have  the 
same  values  for  all  systems  of  any  ensemble.  In  the  case  of 
a  canonical  distribution,  i.  e.,  when  the  index  of  probability 
of  phase  is  a  linear  function  of  the  energy,  it  is  evident  that 
the  values  of  the  external  coordinates  will  affect  the  distribu- 
tion, since  they  affect  the  energy.  In  the  equation 

(105) 

by  which  ty  may  be  determined,  the  external  coordinates,  ax , 
02,  etc.,  contained  implicitly  in  e,  as  well  as  ®,^are  to  be  re- 
garded as  constant  in  the  integrations  indicated.  The  equa- 
tion indicates  that  -fy  is  a  function  of  these  constants.  If  we 
imagine  their  values  varied,  and  the  ensemble  distributed 
canonically  according  to  their  new  values,  we  have  by 
differentiation  of  the  equation  ^ 

/  v  aii 

f     i  ./.       \        1          /» 

0 


,  \ 

(-  I  ^  +  I «»)  =  p 


all 


phases 

all  Jf 

•  •  -/^  e~°  dPi  •  •  •  dv-  ~  ete->      (106) 

phases 
t 

or,  multiplying  by  0  e®,  and  setting 

-^=^   -£=^  etc-> 

all 


|d®  =  ^®  f.  .  .f 


ee 

phases 


— 

i  e  ®   dpl  .  .  .  dqn 

phases 


r      r 

i  I  .  .  . 

phases 

r    *  (•         fcf 

2J  ...JA2e&dpl...dqn    +  etc.       (107) 


44  CANONICAL  DISTRIBUTION 

Now  the  average  value  in  the  ensemble  of  any  quantity 
(which  we  shall  denote  in  general  by  a  horizontal  line  above 
the  proper  symbol)  is  determined  by  the  equation 

r  M  C     fc! 
«  =J  •  •  •  J  u  e  &  dPl...  dqa.  (108) 

phases 

Comparing  this  with  the  preceding  equation,  we  have 

<ty  =  £  d®  -  ~  d®  -  A!  da^  -  22  da2  -  etc.  (109) 

(jj)  (y 

Or,  since  fe—  J  =  ,,  (110) 

and  ^=^ 


d\f/  =  yd®  —  AI  da,i  —  >Z2  d«2  —  etc. 
Moreover,  since  (111)  gives 

dty  -  c?e  =  ©cfy  +  ^©,  (113) 

we  have  also 

dk  —  —  ®  drj  —  ^  ddi  —  A2  da2  —  etc.  (114) 

This  equation,  if  we  neglect  the  sign  of  averages,  is  identi- 
cal in  form  with  the  thermodynamic  equation 

de  +  Alda1  +  Az  daz  +  etc. 
drj=  —y—  -,  (115) 

or 

de  =  Td-rj  —  A!  daL  —  Az  da2  —  etc.,  (H6) 

which  expresses  the  relation  between  the  energy,  .tempera- 
ture, and  entropy  of  a  body  in  thermodynamic  equilibrium, 
and  the  forces  which  it  exerts  on  external  bodies,  —  a  relation 
which  is  the  mathematical  expression  of  the  second  law  of 
thermodynamics  for  reversible  changes.  The  modulus  in  the 
statistical  equation  corresponds  to  temperature  in  the  thermo- 
dynamic equation,  and  the  average  index  of  probability  with 
its  sign  reversed  corresponds  to  entropy.  But  in  the  thermo- 
dynamic equation  the  entropy  (77)  is  a  quantity  which  is 


OF  AN  ENSEMBLE  OF  SYSTEMS.  45 

only  defined  by  the  equation  itself,  and  incompletely  defined 
in  that  the  equation  only  determines  its  differential,  and  the 
constant  of  integration  is  arbitrary.  On  the  other  hand,  the 
77  in  the  statistical  equation  has  been  completely  defined  as 
the  average  value  in  a  canonical  ensemble  of  systems  of 
the  logarithm  of  the  coefficient  of  probability  of  phase. 

We  may  also  compare  equation  (112)  with  the  thermody- 
namic  equation 

A^  =  —  T]dT—Aldal  —  Azda<i  —  etc.,  (117) 

where  ^r  represents  the  function  obtained  by  subtracting  the 
product  of  the  temperature  and  entropy  from  the  energy. 

How  far,  or  in  what  sense,  the  similarity  of  these  equations 
constitutes  any  demonstration  of  the  thermodynamic  equa- 
tions, or  accounts  for  the  behavior  of  material  systems,  as 
described  in  the  theorems  of  thermodynamics,  is  a  question 
of  which  we  shall  postpone  the  consideration  until  we  have 
further  investigated  the  properties  of  an  ensemble  of  systems 
distributed  in  phase  according  to  the  law  which  we  are  con- 
sidering. The  analogies  which  have  been  pointed  out  will  at 
least  supply  the  motive  for  this  investigation,  which  will 
naturally  commence  with  the  determination  of  the  average 
values  in  the  ensemble  of  the  most  important  quantities  relating 
to  the  systems,  and  to  the  distribution  of  the  ensemble  with 
respect  to  the  different  values  of  these  quantities. 


CHAPTER  V. 

AVERAGE  VALUES  IN  A  CANONICAL  ENSEMBLE 
OF  SYSTEMS. 

IN  the  simple  but  important  case  of  a  system  of  material 
points,  if  we  use  rectangular  coordinates,  we  have  for  the 
product  of  the  differentials  of  the  coordinates 

dxi  dyi  dzi  .  .  .  dxv  dyv  dzv, 

and  for  the  product  of  the  differentials  of  the  momenta 
ml  dxi  mi  dyi  m^  dz1  .  .  .  mv  dxv  mv  dyv  mv  dzv . 


The  product  of  these  expressions,  which  represents  an  element 
of  extension-in-phase,  may  be  briefly  written 

mi  dxi  .  .  .  mv  dzv  dxi  .  . .  dzv ; 
and  the  integral 

e  @  mi  dxi  .  . .  mv  dzv  dxi .  .  .  dzv  (118) 

will  represent  the  probability  that  a  system  taken  at  random 
from  an  ensemble  canonically  distributed  will  fall  within  any 
given  limits  of  phase. 
In  this  case 


(119) 
and 


e 


0   =e  &    e     2€>  •••«     20  .  (120) 


The  potential  energy  (e3)  is  independent  of  the  velocities, 
and  if  the  limits  of  integration  for  the  coordinates  are  inde- 
pendent of  the  velocities,  and  the  limits  of  the  several  veloci- 
ties are  independent  of  each  other  as  well  as  of  the  coordinates, 


VALUES  IN  A    CANONICAL  ENSEMBLE.  47 

the   multiple  integral  may  be  resolved  into  the   product   of 
integrals 


C.  .  .  C 


mvdzv.   (121) 


This  shows  that  the  probability  that  the  configuration  lies 
within  any  given  limits  is  independent  of  the  velocities, 
and  that  the  probability  that  any  component  velocity  lies 
within  any  given  limits  is  independent  of  the  other  component 
velocities  and  of  the  configuration. 
Since 

*  2 

f 4V«>,  <&  =  vz^®,         (122> 

I/      00 

and 


J 


e     2  ®  m*  dx!  =  V^Ti-mx©8,  (123> 


the  average  value  of  the  part  of  the  kinetic  energy  due  to  the 
velocity  x19  which  is  expressed  by  the  quotient  of  these  inte- 
grals, is  J  <H).  This  is  true  whether  the  average  is  taken  for 
the  whole  ensemble  or  for  any  particular  configuration, 
whether  it  is  taken  without  reference  to  the  other  component 
velocities,  or  only  those  systems  are  considered  in  which  the 
other  component  velocities  have  particular  values  or  lie 
within  specified  limits. 

The  number  of  coordinates  is  3  v  or  n.     We  have,  therefore, 
for  the  average  value  of  the  kinetic  energy  of  a  system 

ep  =  !„©  =  £  w©.  (124) 


This  is  equally  true  whether  we  take  the  average  for  the  whole 
ensemble,  or  limit  the  average  to  a  single  configuration. 

The  distribution  of  the  systems  with  respect  to  their  com- 
ponent velocities  follows  the  *  law  of  errors  '  ;  the  probability 
that  the  value  of  any  component  velocity  lies  within  any  given 
limits  being  represented  by  the  value  of  the  corresponding 
integral  in  (121)  for  those  limits,  divided  by  (2  TT  m  ®)*, 


48  AVERAGE   VALUES  IN  A    CANONICAL 

which  is  the  value  of  the  same  integral  for  infinite  limits. 
Thus  the  probability  that  the  value  of  x^  lies  between  any 
given  limits  is  expressed  by 


C 
J 


e     2&  dXl.  (125) 


The  expression  becomes  more  simple  when  the  velocity  is 
expressed  with  reference  to  the  energy  involved.  If  we  set 

s=(^xl, 

the  probability  that  s  lies  between  any  given  limits  is 
expressed  by 

~S*ds.  (126) 

Here  s  is  the  ratio  of  the  component  velocity  to  that  which 
would  give  the  energy  ® ;  in  other  words,  s2  is  the  quotient 
of  the  energy  due  to  the  component  velocity  divided  by  ®. 
The  distribution  with  respect  to  the  partial  energies  due  to 
the  component  velocities  is  therefore  the  same  for  all  the  com- 
ponent velocities. 

The  probability  that  the  configuration  lies  within  any  given 
limits  is  expressed  by  the  value  of 

M f  (27r©)¥  f .  .  .  /**.**  .  .  .  dzv  (127) 

for  those  limits,  where  M  denotes  the  product  of  all  the 
masses.  This  is  derived  from  (121)  by  substitution  of  the 
values  of  the  integrals  relating  to  velocities  taken  for  infinite 
limits. 

Very  similar  results  may  be  obtained  in  the  general  case  of 
a  conservative  system  of  n  degrees  of  freedom.  Since  ep  is  a 
homogeneous  quadratic  function  of  the  ^>'s,  it  may  be  divided 
into  parts  by  the  formula 

_    1  ^^p  -I  @£p  /-I  OQ\ 


ENSEMBLE   OF  SYSTEMS.  49 

where  e  might  be  written  for  ep  in  the  differential  coefficients 
without  affecting  the  signification.  The  average  value  of  the 
first  of  these  parts,  for  any  given  configuration,  is  expressed 
by  the  quotient 

/+»  f+»  de    ^r  . 

•  •  •  /  i*l  ~fo     6  dPl  '  '  •   dPn 

_oo  J  —oo  api 

-=r-  (129) 


e  ®   dpi  .  . .  dpn 
Now  we  have  by  integration  by  parts 


ty-C 

r  °°  PI  <^~^-  dPl  =  ©  r  4 

,/  _oo  api  j  _ 


By  substitution  of  this  value,  the  above  quotient  reduces  to 

—  ,  which  is  therefore  the  average  value  of  \P\—  for  the 
2  dpi 

given  configuration.  Since  this  value  is  independent  of  the 
configuration,  it  must  also  be  the  average  for  the  whole 
ensemble,  as  might  easily  be  proved  directly.  (To  make 
the  preceding  proof  apply  directly  to  the  whole  ensemble,  we 
have  only  to  write  dp1  .  .  .  dqn  for  dp±  .  .  .  dpn  in  the  multiple 
integrals.)  This  gives  J  n  ®  for  the  average  value  of  the 
whole  kinetic  energy  for  any  given  configuration,  or  for 
the  whole  ensemble,  as  has  already  been  proved  in  the  case  of 
material  points. 

The  mechanical  significance  of  the  several  parts  into  which 
the  kinetic  energy  is  divided  in  equation  (128)  will  be  appar- 
ent if  we  imagine  that  by  the  application  of  suitable  forces 
(different  from  those  derived  from  eq  and  so  much  greater 
that  the  latter  may  be  neglected  in  comparison)  the  system 
was  brought  from  rest  to  the  state  of  motion  considered,  so 
rapidly  that  the  configuration  was  not  sensibly  altered  during 
the  process,  and  in  such  a  manner  also  that  the  ratios  of  the 
component  velocities  were  constant  in  the  process.  If  we 
write 


50  AVERAGE    VALUES  IN  A    CANONICAL 

for  the  moment  of  these  forces,  we  have  for  the  period  of  their 
action  by  equation  (3) 

*  =-(^-d^  +  Fl  =  -  —  +  Fl 

dqi       dqi  dqi 

The  work  done  by  the  force  F±  may  be  evaluated  as  follows : 

r         rd€  * 

=  I  Pi  dqt  -f    I  y—dqit 
J  J  dq^ 

where  the  last  term  may  be  cancelled  because  the  configuration 
does  not  vary  sensibly  during  the  application  of  the  forces. 
(It  will  be  observed  that  the  other  terms  contain  factors  which 
increase  as  the  tune  of  the  action  of  the  forces  is  diminished.) 
We  have  therefore, 

f*  f*  n      f* 

\  dqi  =  I  pi  £1  dt  =  I  qi  dpt=.  —  I  Pi  dpi .          (131) 

For  since  the  p's  are  linear  functions  of  the  q's  (with  coeffi- 
cients involving  the  #'s)  the  supposed  constancy  of  the  <?'s  and 
of  the  ratios  of  the  <?'s  will  make  the  ratio  fa/Pi  constant. 
The  last  integral  is  evidently  to  be  taken  between  the  limits 
zero  and  the  value  of  p1  in  the  phase  originally  considered, 
and  the  quantities  before  the  integral  sign  may  be  taken  as 
relating  to  that  phase.  We  have  therefore 

i  =  ipl^Lt  (132) 

That  is:  the  several  parts  into  which  the  kinetic  energy  is 
divided  in  equation  (128)  represent  the  amounts  of  energy 
communicated  to  the  system  by  the  several  forces  Fl ,  .  .  .  Fn 
under  the  conditions  mentioned. 

The  following  transformation  will  not  only  give  the  value 
of  the  average  kinetic  energy,  but  will  also  serve  to  separate 
the  distribution  of  the  ensemble  in  configuration  from  its  dis- 
tribution in  velocity. 

Since  2  ep  is  a  homogeneous  quadratic  function  of  the  jo's, 
which  is  incapable  of  a  negative  value,  it  can  always  be  ex- 
pressed (and  in  more  than  one  way)  as  a  sum  of  squares  of 


» 


ENSEMBLE   OF  SYSTEMS.  51 

linear  functions  of  the  JD'S.*  The  coefficients  in  these  linear 
functions,  like  those  in  the  quadratic  function,  must  be  regarded 
in  the  general  case  as  functions  of  the  <?'s.  Let 

2ep  =  <2  +  w22...  +  iv2  (133) 

where  MJ  .  .  .  un  are  such  linear  functions  of  the  p'a.  If  we 
write 


for  the  Jacobian  or  determinant  of  the  differential  coefficients 
of  the  form  dp/du,  we  may  substitute 


for  dp1  .  .  .  dpn 

under  the  multiple  integral  sign  in  any  of  our  formulae.  It 
will  be  observed  that  this  determinant  is  function  of  the  <?'s 
alone.  The  sign  of  such  a  determinant  depends  on  the  rela- 
tive order  of  the  variables  in  the  numerator  and  denominator. 
But  since  the  suffixes  of  the  it's  are  only  used  to  distinguish 
these  functions  from  one  another,  and  no  especial  relation  is 
supposed  between  a  p  and  a  u  which  have  the  same  suffix,  we 
may  evidently,  without  loss  of  generality,  suppose  the  suffixes 
so  applied  that  the  determinant  is  positive. 

Since  the  w's  are  linear  functions  of  the  />'s,  when  the  in- 
tegrations are  to  cover  all  values  of  the  jt?'s  (for  constant  #'s) 
once  and  only  once,  they  must  cover  all  values  of  the  w's  once 
and  only  once,  and  the  limits  will  be  ±  oo  for  all  the  u's. 
Without  the  supposition  of  the  last  paragraph  the  upper  limits 
would  not  always  be  +  oo  ,  as  is  evident  on  considering  the 
effect  of  changing  the  sign  of  a  u.  But  with  the  supposition 
which  we  have  made  (that  the  determinant  is  always  positive) 
we  may  make  the  upper  limits  +  oo  and  the  lower  —  oo  for  all 
the  t*'s.  Analogous  considerations  will  apply  where  the  in- 
tegrations do  not  cover  all  values  of  the  p's  and  therefore  of 

*  The  reduction  requires  only  the  repeated  application  of  the  process  of 
'completing  the  square*  used  in  the  solution  of  quadratic  equations. 


52  AVERAGE    VALUES  IN  A    CANONICAL 

the  w's.     The  integrals  may  always  be  taken  from  a  less  to  a 
greater  value  of  a  u. 

The  general  integral  which  expresses  the  fractional  part  of 
the  ensemble  which  falls  within  any  given  limits  of  phase  is 
thus  reduced  to  the  form 


...<*«*«*&...%,.  (134) 

For  the  average  value  of  the  part  of  the  kinetic  energy 
which  is  represented  by  ^u^  whether  the  average  is  taken 
for  the  whole  ensemble,  or  for  a  given  configuration,  we  have 
therefore 


__         (135) 

—  --' 


I/ 


e 

00 


and  for  the  average  of  the  whole  kinetic  energy,  JTI©,  as 
before. 

The  fractional  part  of  the  ensemble  which  lies  within  any 
given  limits  of  configuration,  is  found  by  integrating  (184) 
with  respect  to  the  w's  from  —  oo  to  +  oo  .  This  gives 


J  f. 


•    da, 


which  shows  that  the  value  of  the  Jacobian  is  independent  of 
the  manner  in  which  2ep  is  divided  into  a  sum  of  squares. 
We  may  verify  this  directly,  and  at  the  same  tune  obtain  a 
more  convenient  expression  for  the  Jacobian,  as  follows. 

It  will  be  observed  that  since  the  M'S  are  linear  functions  of 
the  p's,  and  the  jt?'s  linear  functions  of  the  ^'s,  the  u's  will  be 
linear  functions  of  the  <?'s,  so  that  a  differential  coefficient  of 
the  form  du/dq  will  be  independent  of  the  q's,  and  function  of 
the  <?'s  alone.  Let  us  write  dpxjduy  for  the  general  element 
of  the  Jacobian  determinant.  We  have 


ENSEMBLE   OF  SYSTEMS.  53 

dpx         d    de  d    r=n  de    dur 

duy       duy  dqx       duy  r—\  dur  dqx 

— r?"  (     ^e      dur\         d     de   _  duy 

Therefore 

d(p,  ...pn)  __d(u,  ..  .  Q 
d(u,  .  . .  u^)       d(q,  .  .  .  qn) 

and 

^«.  ^) 


These  determinants  are  all  functions  of  the  <?'s  alone.*  The 
last  is  evidently  the  Hessian  or  determinant  formed  of  the 
second  differential  coefficients  of  the  kinetic  energy  with  re- 
spect to  <?j ,  . . .  qn.  We  shall  denote  it  by  Aj.  The  reciprocal 
determinant 


which  is  the  Hessian  of  the  kinetic  energy  regarded  as  func- 
tion of  the  p's,  we  shall  denote  by  Ap. 
If  we  set 

e  &  =  I  .  . .  /    e  ®  Ap    dp,...dpn 


+00  +00    —Mj2  .  .  .  —  «n2 


f.  .  .  C 


e       20        dUl  .  .  .  dun  =  (27r©)§,         (140) 


and  *,  =  *  -  fe  (141) 

*  It  will  be  observed  that  the  proof  of  (137)  depends  on  the  linear  relation 

dur 

between  the  u's  and  q's,  which  makes  —  —  constant  with  respect  to  the  differ- 

dqx 

entiations  here  considered.    Compare  note  on  p.  12. 


54  AVERAGE    VALUES  IN  A    CANONICAL 

the  fractional  part  of  the  ensemble  which  lies  within  any 
given  limits  of  configuration  (136)  may  be  written 

•   dql .  .  .  dqn,  (142) 

where  the  constant  tyq  may  be  determined  by  the  condition 
that  the  integral  extended  over  all  configurations  has  the  value 
unity.* 

*  In  the  simple  but  important  case  in  which  Aj  is  independent  of  the  ^'s, 
and  €j  a  quadratic  function  of  the  q's,  if  we  write  ea  for  the  least  value  of  €q 
(or  of  e)  consistent  with  the  given  values  of  the  external  coordinates,  the 
equation  determining  \l/q  may  be  written 


—  00         00 

If  we  denote  by  q±t . . .  qn'  the  values  of  qi , . . .  qn  which  give  fq  its  least  value 
ea ,  it  is  evident  that  eg  —  ea  is  a  homogenous  quadratic  function  of  the  differ- 
ences ?!  —  qi,  etc.,  and  that  dqt, . . .  dqn  may  be  regarded  as  the  differentials 
of  these  differences.  The  evaluation  of  this  integral  is  therefore  analytically 
similar  to  that  of  the  integral 

+00        +00_J 

J.  .  .fe     &  dp!  . .  .  dpn, 

00          —CO 

for  which  we  have  found  the  value  Ap *  (2  TT  9)  3.  By  the  same  method,  or 
by  analogy,  we  get 


where  A9  is  the  Hessian  of  the  potential  energy  as  function  of  the  q's.  It 
will  be  observed  that  A?  depends  on  the  forces  of  the  system  and  is  independ- 
ent of  the  masses,  while  A^  or  its  reciprocal  Ap  depends  on  the  masses  and 
is  independent  of  the  forces.  While  each  Hessian  depends  on  the  system  of 
coordinates  employed,  the  ratio  A^/A^  is  the  same  for  all  systems. 
Multiplying  the  last  equation  by  (140),  we  have 


For  the  average  value  of  the  potential  energy,  we  have 

+00      +00  *g~ea 

J  '  '  -f  (€Q  —  fa)e  dql  .  .  .  dqn 


—00          —00 


+00       +eo * a 

J  .  .  .J  e  dqi .  . .  dqn 


ENSEMBLE  OF  SYSTEMS.  55 

When  an  ensemble  of  systems  is  distributed  in  configura- 
tion in  the  manner  indicated  in  this  formula,  i.  e.,  when  its 
distribution  in  configuration  is  the  same  as  that  of  an  en- 
semble canonically  distributed  in  phase,  we  shall  say,  without 
any  reference  to  its  velocities,  that  it  is  canonically  distributed 
in  configuration. 

For  any  given  configuration,  the  fractional  part  of  the 
systems  which  lie  within  any  given  limits  of  velocity  is 
represented  by  the  quotient  of  the  multiple  integral 


®dPl...dpn, 
or  its  equivalent 


- 


l-- 


taken  within  those  limits  divided  by  the  value  of  the  same 
integral  for  the  limits  ±  oo.  But  the  value,  of  the  second 
multiple  integral  for  the  limits  ±  oo  is  evidently 


We  may  therefore  write 

~~®~  du^  .  . .  dun,  (143) 


The  evaluation  of  this  expression  is  similar  to  that  of 

+00          +00         _!? 

...sfe      &dpl...dpn 


+00      +00  _CJL 
f...fe      &dPl...dpn 

-  00         -  CO 

which  expresses  the  average  value  of  the  kinetic  energy,  and  for  which  we 
have  found  the  value  $  n  6.    We  have  accordingly 

«4-«a  =  2na 
Adding  the  equation 

*i>  =  2ne> 
we  have  I  —  ea  =  n  e. 


\ 


\ 


56  AVERAGES  IN  A    CANONICAL  ENSEMBLE. 

/„  ^p-fp 
•••je    &    **dPl...dpn,  (144) 


or  again 


r      r^=^    i 

I  .  .  .  /  e   <     Ar^Ti  •  •  •  4»i  (145) 

for  the  fractional  part  of  the  systems  of  any  given  configura- 
tion which  lie  within  given  limits  of  velocity. 

When  systems  are  distributed  in  velocity  according  to  these 
formulae,  i.  e.,  when  the  distribution  in  velocity  is  like  that  in 
an  ensemble  which  is  canonically  distributed  in  phase,  we 
shall  say  that  they  are  canonically  distributed  in  velocity. 

The  fractional  part  of  the  whole  ensemble  which  falls 
within  any  given  limits  of  phase,  which  we  have  before 
expressed  in  the  form 


.  dpndqi  .  .  .  dqn,  (146) 

may  also  be  expressed  in  the  form 

.  .  dqndql  .  .  .  dqn.  (147) 


CHAPTER  VI. 

EXTENSION  IN  CONFIGURATION  AND  EXTENSION 
IN  VELOCITY. 

THE  formulae  relating  to  canonical  ensembles  in  the  closing 
paragraphs  of  the  last  chapter  suggest  certain  general  notions 
and  principles,  which  we  shall  consider  in  this  chapter,  and 
which  are  not  at  all  limited  in  their  application  to  the  canon- 
ical law  of  distribution.* 

We  have  seen  in  Chapter  IV.  that  the  nature  of  the  distribu- 
tion which  we  have  called  canonical  is  independent  of  the 
system  of  coordinates  by  which  it  is  described,  being  deter- 
mined entirely  by  the  modulus.  It  follows  that  the  value 
represented  by  the  multiple  integral  (142),  which  is  the  frac- 
tional part  of  the  ensemble  which  lies  within  certain  limiting 
configurations,  is  independent  of  the  system  of  coordinates, 
being  determined  entirely  by  the  limiting  configurations  with 
the  modulus.  Now  t|r,  as  we  have  already  seen,  represents 
a  value  which  is  independent  of  the  system  of  coordinates 
by  which  it  is  defined.  The  same  is  evidently  true  of 
typ  by  equation  (140),  and  therefore,  by  (141),  of  tyg. 
Hence  the  exponential  factor  in  the  multiple  integral  (142) 
represents  a  value  which  is  independent  of  the  system  of 
coordinates.  It  follows  that  the  value  of  a  multiple  integral 
of  the  form 

^  ...dgn  (148) 


*  These  notions  and  principles  are  in  fact  such  as  a  more  logical  arrange- 
ment of  the  subject  would  place  in  connection  with  those  of  Chapter  I.,  to 
which  they  are  closely  related.  The  strict  requirements  of  logical  order 
have  been  sacrificed  to  the  natural  development  of  the  subject,  and  very 
elementary  notions  have  been  left  until  they  have  presented  themselves  in 
the  study  of  the  leading  problems. 


58  EXTENSION  IN  CONFIGURATION 

is  independent  of  the  system  of  coordinates  which  is  employed 
for  its  evaluation,  as  will  appear  at  once,  if  we  suppose  the 
multiple  integral  to  be  broken  up  into  parts  so  small  that 
the  exponential  factor  may  be  regarded  as  constant  in  each. 
In  the  same  way  the  formulae  (144)  and  (145)  which  express 
the  probability  that  a  system  (in  a  canonical  ensemble)  of  given 
configuration  will  fall  within  certain  limits  of  velocity,  show 
that  multiple  integrals  of  the  form 


(149) 


or  *»     **&„.  1*  (150) 

relating  to  velocities  possible  for  a  given  configuration,  when 
the  limits  are  formed  by  given  velocities,  have  values  inde- 
pendent of  the  system  of  coordinates  employed. 

These  relations  may  easily  be  verified  directly.     It  has  al- 
ready been  proved  that 

d(Pl9  .  .  .  P.)        <%i  .  .  .  qn)          d(ql9  ...qn) 


..-)      d(Ql9...Qn) 

where  ql  ,  .  .  .  q^ft  ,  .  .  .pn  and  Ql  ,  .  .  .  Qn9  P1  ,  .  .  .  Pn  are  two 
systems  of  coordinates  and  momenta.*    It  follows  that 


i> 


=  r 

J 


*  See  equation  (29). 


AND  EXTENSION  IN   VELOCITY.  59 

and 

/Cfd(Ql,  ...  Qn)\%  JT>  Jp 

'  '  J  \d(P^   ~^P})  '  * 

"'<%>!...  <jp. 


=  c. 

J 


>!,-..  W 

The  multiple  integral 
• 

>!  .  .  .  dpndqi  .  .  .  rf^,  (151) 

which  may  also  be  written 

£1  .  .  .  dqndqi  .  .  .  dqn,  (152) 


and  which,  when  taken  within  any  given  limits  of  phase,  has 
been  shown  to  have  a  value  independent  of  the  coordinates 
employed,  expresses  what  we  have  called  an  extension-in- 
phase.*  In  like  manner  we  may  say  that  the  multiple  integral 
(148)  expresses  an  extension-in-configuration,  and  that  the 
multiple  integrals  (149)  and  (150)  express  an  extensionrin- 
velocity.  We  have  called 

dpi  .  .  .  <Zp.<fyi  .  .  .  dqn,  (153) 

which  is  equivalent  to 

A-^!  .  .  .  dqndqt  .  .  .  dqn,  (154) 

an  element  of  extension-in-phase.     We  may  call 

A^  ...dqn  (155) 

an  element  of  extension-in-configuration,  and 

.  .  .  dpn,  (156) 


See  Chapter  I,  p.  10. 


60  EXTENSION  IN  CONFIGURATION 

or  its  equivalent 


.  .  d,  (157) 

an  element  of  extension-in-velocity. 

An  extension-in-phase  may  always  be  regarded  as  an  integral 
of  elementary  extensions-in-configuration  multiplied  each  by 
an  extension-in-velocity.  This  is  evident  from  the  formulae 
(151)  and  (152)  which  express  an  extension-in-phase,  if  we 
imagine  the  integrations  relative  to  velocity  to  be  first  carried 
out. 

The  product  of  the  two  expressions  for  an  element  of 
extension-in-velocity  (149)  and  (150)  is  evidently  of  the  same 
dimensions  as  the  product 

Pi-  '  -PnVl  •   --it 

that  is,  as  the  nth  power  of  energy,  since  every  product  of  the 
form  pl  q1  has  the  dimensions  of  energy.  Therefore  an  exten- 
sion-in-velocity has  the  dimensions  of  the  square  root  of  the 
nth  power  of  energy.  Again  we  see  by  (155)  and  (156)  that 
the  product  of  an  extension-in-configuration  and  an  extension- 
in-velocity  have  the  dimensions  of  the  nth  power  of  energy 
multiplied  by  the  nth  power  of  time.  Therefore  an  extension- 
in-configuration  has  the  dimensions  of  the  nth  power  of  time 
multiplied  by  the  square  root  of  the  nth  power  of  energy. 

To  the  notion  of  extension-in-configuration  there  attach 
themselves  certain  other  notions  analogous  to  those  which  have 
presented  themselves  in  connection  with  the  notion  of  ex- 
tension-in-phase. The  number  of  systems  of  any  ensemble 
(whether  distributed  canonically  or  in  any  other  manner) 
which  are  contained  in  an  element  of  extension-in-configura- 
tion, divided  by  the  numerical  value  of  that  element,  may  be 
called  the  density-in-configuration.  That  is,  if  a  certain  con- 
figuration is  specified  by  the  coordinates  q1  .  .  .  qn,  and  the 
number  of  systems  of  which  the  coordinates  fall  between  the 
limits  q1  and  ql  +  dql  ,  .  .  .  qn  and  qn  +  dqn  is  expressed  by 

D.A^Zi  •  •  •  *2n,  (158) 


AND  EXTENSION  IN  VELOCITY.  61 

Dq  will  be  the  density-in-configuration.     And  if  we  set 

«*=ip  (159) 

where  N  denotes,  as  usual,  the  total  number  of  systems  in  the 
ensemble,  the  probability  that  an  unspecified  system  of  the 
ensemble  will  fall  within  the  given  limits  of  configuration,  is 
expressed  by 

e^dqt  .  .  .  dqn.  (160) 

We  may  call  &*  the  coefficient  of  probability  of  the,  configura- 
tion, and  t]q  the  index  of  probability  of  the  configuration. 

The  fractional  part  of  the  whole  number  of  systems  which 
are  within  any  given  limits  of  configuration  will  be  expressed 
by  the  multiple  integral 


J. 


.  .  .  dgn.  (161) 


The  value  of  this  integral  (taken  within  any  given  configura- 
tions) is  therefore  independent  of  the  system  of  coordinates 
which  is  used.  Since  the  same  has  been  proved  of  the  same 
integral  without  the  factor  e*q,  it  follows  that  the  values  of 
7)q  and  Dq  for  a  given  configuration  in  a  given  ensemble  are 
independent  of  the  system  of  coordinates  which  is  used. 

The  notion  of  extension-in-velocity  relates  to  systems  hav- 
ing the  same  configuration.*  If  an  ensemble  is  distributed 
both  in  configuration  and  in  velocity,  we  may  confine  our 
attention  to  those  systems  which  are  contained  within  certain 
infinitesimal  limits  of  configuration,  and  compare  the  whole 
number  of  such  systems  with  those  which  are  also  contained 

*  Except  in  some  simple  cases,  such  as  a  system  of  material  points,  we 
cannot  compare  velocities  in  one  configuration  with  velocities  in  another,  and 
speak  of  their  identity  or  difference  except  in  a  sense  entirely  artificial.  We 
may  indeed  say  that  we  call  the  velocities  in  one  configuration  the  same  as 
those  in  another  when  the  quantities  qlt  ...qn  have  the  same  values  in  the 
two  cases.  But  this  signifies  nothing  until  the  system  of  coordinates  has 
been  defined.  We  might  identify  the  velocities  in  the  two  cases  which  make 
the  quantities  pi,...pn  the  same  in  each.  This  again  would  signify  nothing 
independently  of  the  system  of  coordinates  employed. 


62  EXTENSION  IN  CONFIGURATION 

within  certain  infinitesimal  limits  of  velocity.  The  second 
of  these  numbers  divided  by  the  first  expresses  the  probability 
that  a  system  which  is  only  specified  as  falling  within  the  in- 
finitesimal limits  of  configuration  shall  also  fall  within  the 
infinitesimal  limits  of  velocity.  If  the  limits  with  respect  to 
velocity  are  expressed  by  the  condition  that  the  momenta 
shall  fall  between  the  limits  p1  and  p1  +  dpl  ,  .  .  .  pn  and 
Pn  +  dpm  the  extension-in-velocity  within  those  limits  will  be 


.  .  .  dpn, 
and  we  may  express  the  probability  in  question  by 

e^\^dPl  .  .  .  dpn.  (162) 

This  may  be  regarded  as  defining  rjp  . 

The  probability  that  a  system  which  is  only  specified  as 
having  a  configuration  within  certain  infinitesimal  limits  shall 
also  fall  within  any  given  limits  of  velocity  will  be  expressed 
by  the  multiple  integral 


h  .  .  .  dpn,  (163) 

or  its  equivalent 

J1.  .  .J*»*4Mb  .  .  .  dgn,  (164) 

taken  within  the  given  limits. 

It  follows  that  the  probability  that  the  system  will  fall 
within  the  limits  of  velocity,  ^  and  ^  +  dq19  .  .  .  qn  and 
2»  +  dq*  is  expressed  by 

e^^d^^.d^.  (165) 

The  value  of  the  integrals  (163),  (164)  is  independent  of 
the  system  of  coordinates  and  momenta  which  is  used,  as  is 
also  the  value  of  the  same  integrals  without  the  factor 
e1?;  therefore  the  value  of  TJP  must  be  independent  of  the 
system  of  coordinates  and  momenta.  We  may  call  e1?  the 
coefficient  of  probability  of  velocity,  and  tjp  the  index  of  proba- 
bility of  velocity. 


AND  EXTENSION  IN   VELOCITY.  63 

Comparing  (160)  and  (162)  with  (40),  we  get 

eV*  =  P  =  el  (166) 

or  rjq  +  IP  =  ^.  (167) 

That  is :  the  product  of  the  coefficients  of  probability  of  con- 
figuration and  of  velocity  is  equal  to  the  coefficient  of  proba- 
bility of  phase;  the  sum  of  the  indices  of  probability  of 
configuration  and  of  velocity  is  equal  to  the  index  of 
probability  of  phase. 

It  is  evident  that  e1*  and  e1?  have  the  dimensions  of  the 
reciprocals  of  extension-in-configuration  and  extension-in- 
velocity  respectively,  i.  e.,  the  dimensions  of  t~n  e~*  and  e~», 
where  t  represent  any  tune,  and  e  any  energy.  If,  therefore, 
the  unit  of  time  is  multiplied  by  ct,  and  the  unit  of  energy  by 
ce ,  every  rjq  will  be  increased  by  the  addition  of 

n  log  ct  +  i?i  log  c. ,  (168) 

and  every  rjp  by  the  addition  of 

in  logo.*  (169) 

It  should  be  observed  that  the  quantities  which  have  been 
called  extension-in-configuration  and  extension-in-velocity  are 
not,  as  the  terms  might  seem  to  imply,  purely  geometrical  or 
kinematical  conceptions.  To  express  their  nature  more  fully, 
they  might  appropriately  have  been  called,  respectively,  the 
dynamical  measure  of  the  extension  in  configuration,  and  the 
dynamical  measure  of  the  extension  in  velocity.  They  depend 
upon  the  masses,  although  not  upon  the  forces  of  the 
system.  In  the  simple  case  of  material  points,  where  each 
point  is  limited  to  a  given  space,  the  extension-in-configuration 
is  the  product  of  the  volumes  within  which  the  several  points 
are  confined  (these  may  be  the  same  or  different),  multiplied 
by  the  square  root  of  the  cube  of  the  product  of  the  masses  of 
the  several  points.  The  extension-in-velocity  for  such  systems 
is  most  easily  defined  as  the  extension-in-configuration  of 
systems  which  have  moved  from  the  same  configuration  for 
the  unit  of  time  with  the  given  velocities. 
*  Compare  (47)  in  Chapter  I. 


64  EXTENSION  IN  CONFIGURATION 

In  the  general  case,  the  notions  of  extension-in-configuration 
and  extension-in-velocity  may  be  connected  as  follows. 

If  an  ensemble  of  similar  systems  of  n  degrees  of  freedom 
have  the  same  configuration  at  a  given  instant,  but  are  distrib- 
uted throughout  any  finite  extension-in-velocity,  the  same 
ensemble  after  an  infinitesimal  interval  of  time  St  will  be 
distributed  throughout  an  extension  in  configuration  equal  to 
its  original  extension-in-velocity  multiplied  by  $tn. 

In  demonstrating  this  theorem,  we  shall  write  q^  .  .  .  qnf  for 
the  initial  values  of  the  coordinates.  The  final  values  will 
evidently  be  connected  with  the  initial  by  the  equations 


Now  the  original  extension-in-velocity  is  by  definition  repre- 
sented by  the  integral 

J.  .  ,JV4i  •  -  •  <&,  (171) 

where  the  limits  may  be  expressed  by  an  equation  of  the  form 
F(jll...^)  =  Q.  (172) 

The  same  integral  multiplied  by  the  constant   St*  may  be 
written 

J.  .  .  jVd&ft),  .  .  .  %„&),  (173) 

and  the  limits  may  be  written 


(It  will  be  observed  that  St  as  well  as  A^  is  constant  in  the 
integrations.)     Now  this  integral  is  identically  equal  to 

f.  .  ./A,*  d(q,  -  <?/)  .  .  .  d(q,  .  .  .  ft,'),  (175) 

or  its  equivalent 

AM.  •  •  •  *»  (176) 


f.  •  -/ 


with  limits  expressed  by  the  equation 

/  (ft  -<?/,•••  2.-  2,.')  =0.  (177) 


AND  EXTENSION  IN   VELOCITY.  65 

But  the  systems  which  initially  had  velocities  satisfying  the 
equation  (172)  will  after  the  interval  Bt  have  configurations 
satisfying  equation  (177).  Therefore  the  extension-in-con- 
figuration  represented  by  the  last  integral  is  that  which 
belongs  to  the  systems  which  originally  had  the  extension-in- 
velocity  represented  by  the  integral  (171). 

Since  the  quantities  which  we  have  called  extensions-in- 
phase,  extensions-in-configuration,  and  extensions-in-velocity 
are  independent  of  the  nature  of  the  system  of  coordinates 
used  in  their  definitions,  it  is  natural  to  seek  definitions  which 
shall  be  independent  of  the  use  of  any  coordinates.  It  will  be 
sufficient  to  give  the  following  definitions  without  formal  proof 
of  their  equivalence  with  those  given  above,  since  they  are 
less  convenient  for  use  than  those  founded  on  systems  of  co- 
ordinates, and  since  we  shall  in  fact  have  no  occasion  to  use 
them. 

We  commence  with  the  definition  of  extension-in- velocity. 
We  may  imagine  n  independent  velocities,  Vl , . . .  Vn  of  which  a 
system  in  a  given  configuration  is  capable.  We  may  conceive 
of  the  system  as  having  a  certain  velocity  F~0  combined  with  a 
part  of  each  of  these  velocities  Vl . . .  Vn.  By  a  part  of  V\  is 
meant  a  velocity  of  the  same  nature  as  V\  but  in  amount  being 
anything  between  zero  and  Vr  Now  all  the  velocities  which 
may  be  thus  described  may  be  regarded  as  forming  or  lying  in 
a  certain  extension  of  which  we  desire  a  measure.  The  case 
is  greatly  simplified  if  we  suppose  that  certain  relations  exist 
between  the  velocities  V\  , . . .  Vw  viz  :  that  the  kinetic  energy 
due  to  any  two  of  these  velocities  combined  is  the  sum  of  the 
kinetic  energies  due  to  the  velocities  separately.  In  this  case 
the  extension-in-motion  is  the  square  root  of  the  product  of 
the  doubled  kinetic  energies  due  to  the  n  velocities  Fi , . . .  Vn 
taken  separately. 

The  more  general  case  may  be  reduced  to  this  simpler  case 
as  follows.  The  velocity  F2  may  always  be  regarded  as 
composed  of  two  velocities  Vj  and  V2",  of  which  VJ  is  of 
the  same  nature  as  Vl ,  (it  may  be  more  or  less  in  amount,  or 
opposite  in  sign,)  while  V2"  satisfies  the  relation  that  the 

5 


66  EXTENSION  IN  CONFIGURATION 

kinetic  energy  due  to  Vl  and  V2n  combined  is  the  sum  of  the 
kinetic  energies  due  to  these  velocities  taken  separately.  And 
the  velocity  VB  may  be  regarded  as  compounded  of  three, 

*Y»  F3">  *Y"»  of  which  v*  is  of  the  same  nature  as  Fi  '  V* 
of  the  same  nature  as   V2",  while  VB"f  satisfies  the  relations 

that  if  combined  either  with  Fi  or  V£  the  kinetic  energy  of 
the  combined  velocities  is  the  sum  of  the  kinetic  energies  of 
the  velocities  taken  separately.  When  all  the  velocities 
Fg  ,  .  .  .  Vn  have  been  thus  decomposed,  the  square  root  of  the 
product  of  the  doubled  kinetic  energies  of  the  several  velocities 
PI>  JY'»  JY"»  ete*'  ^H  be  the  value  of  the  extension-in- 
velocity  which  is  sought. 

This  method  of  evaluation  of  the  extension-in-  velocity  which 
we  are  considering  is  perhaps  the  most  simple  and  natural,  but 
the  result  may  be  expressed  in  a  more  symmetrical  form.  Let 
us  write  e12  for  the  kinetic  energy  of  the  velocities  Fx  and  V% 
combined,  diminished  by  the  sum  of  the  kinetic  energies  due 
to  the  same  velocities  taken  separately.  This  may  be  called 
the  mutual  energy  of  the  velocities  V\  and  F2  .  Let  the 
mutual  energy  of  every  pair  of  the  velocities  Fj  ,  .  .  .  Vn  be 
expressed  in  the  same  way.  Analogy  would  make  en  represent 
the  energy  of  twice  V1  diminished  by  twice  the  energy  of  Fi  , 
i.  e.y  en  would  represent  twice  the  energy  of  Fi  ,  although  the 
term  mutual  energy  is  hardly  appropriate  to  this  case.  At  all 
events,  let  en  have  this  signification,  and  e22  represent  twice 
the  energy  of  F^,  etc.  The  square  root  of  the  determinant 

n  €12  ...  €i 


represents  the  value  of  the  extension-in-velocity  determined  as 
above  described  by  the  velocities  V\  ,  .  .  .  FJ,. 

The  statements  of  the  preceding  paragraph  may  be  readily 
proved  from  the  expression  (157)  on  page  60,  viz., 


A  • 


by  which  the  notion  of  an  element  of  extension-in-velocity  was 


AND  EXTENSION  IN  VELOCITY.  67 

originally  defined.     Since   A^  in  this  expression   represents 
the  determinant  of  which  the  general  element  is 


the  square  of  the  preceding  expression  represents  the  determi- 
nant of  which  the  general  element  is 


Now  we  may  regard  the  differentials  of  velocity  dqt,  d^  as 
themselves  infinitesimal  velocities.  Then  the  last  expression 
represents  the  mutual  energy  of  these  velocities,  and 

d*e 


represents  twice  the  energy  due  to  the  velocity  dq{. 

The  case  which  we  have  considered  is  an  extension-in-veloc- 
ity  of  the  simplest  form.  All  extensions-in-velocity  do  not 
have  this  form,  but  all  may  be  regarded  as  composed  of 
elementary  extensions  of  this  form,  in  the  same  manner  as 
all  volumes  may  be  regarded  as  composed  of  elementary 
parallelepipeds. 

Having  thus  a  measure  of  extension-in-  velocity  founded,  it 
will  be  observed,  on  the  dynamical  notion  of  kinetic  energy, 
and  not  involving  an  explicit  mention  of  coordinates,  we  may 
derive  from  it  a  measure  of  extension-in-configuration  by  the 
principle  connecting  these  quantities  which  has  been  given  in 
a  preceding  paragraph  of  this  chapter. 

The  measure  of  extension-in-phase  may  be  obtained  from 
that  of  extension-in-configuration  and  of  extension-in-  velocity. 
For  to  every  configuration  in  an  extension-in-phase  there  will 
belong  a  certain  extension-in-velocity,  and  the  integral  of  the 
elements  of  extension-in-configuration  within  any  extension- 
in-phase  multiplied  each  by  its  extension-in-velocity  is  the 
measure  of  the  extension-in-phase. 


CHAPTER  VII. 

FARTHER    DISCUSSION    OF  AVERAGES    IN  A  CANONICAL 
ENSEMBLE  OF  SYSTEMS. 

RETURNING  to  the  case  of  a  canonical  distribution,  we  have 
for  the  index  of  probability  of  configuration 


as  appears  on  comparison  of  formulae  (142)  and  (161).  It 
follows  immediately  from  (142)  that  the  average  value  in  the 
ensemble  of  any  quantity  u  which  depends  on  the  configura- 
tion alone  is  given  by  the  formula 

r  au  ^     *<r-*g 
=J...Jue    "    ^dqi...dqn}  (179) 


u 

conflg. 


where  the  integrations  cover  all  possible  configurations.     The 
value  of  i|rg  is  evidently  determined  by  the  equation 

r  ^  r  _!? 

=J  .  .  .J  e   %*dfc  .  .  .  dqn.  (180) 


e 

config. 


By  differentiating  the  last  equation  we  may  obtain  results 
analogous  to  those  obtained  in  Chapter  IV  from  the  equation 


£      -  *"  ~    f 

0 


J       *    J  &    &dPl  '  '  ' 


e 

«. 

phases 


As  the  process  is  identical,  it  is  sufficient  to  give  the  results : 
dfa  =  rjqd®  —  J^i  —  J^da^  —  etc.,  (181) 


AVERAGES  IN  A    CANONICAL  ENSEMBLE.          69 
or,  since  \f/q  =  7g  +  ®^g,  (182) 

and  <fyc  =  <£g  4-  ^grf®  +  ®<fya,  (183) 

ckg  =  —  ©cfyg  —  ^etai  —  J2^«2  —  etc.  (184) 

It  appears  from  this  equation  that  the  differential  relations 
subsisting  between  the  average  potential  energy  in  an  ensem- 
ble of  systems  canonically  distributed,  the  modulus  of  distri- 
bution, the  average  index  of  probability  of  configuration,  taken 
negatively,  and  the  average  forces  exerted  on  external  bodies, 
are  equivalent  to  those  enunciated  by  Clausius  for  the  potential 
energy  of  a  body,  its  temperature,  a  quantity  which  he  called 
the  disgregation,  and  the  forces  exerted  on  external  bodies.* 

For  the  index  of  probability  of  velocity,  in  the  case  of  ca- 
nonical distribution,  we  have  by  comparison  of  (144)  and  (163), 
or  of  (145)  and  (164), 

(185) 

which  gives  ^  =  Yp  ~  *p  ;  (186) 

we  have  also                         ^,  =  £  n  ®,  (187) 

and  by  (140),  fa  =  -  \  n  ©  log  (2ir0).  (188) 
From  these  equations  we  get  by  differentiation 

<%=^d®,  (189) 

and                                       <£,  =  —  ®  d^.  (190) 

The  differential  relation  expressed  in  this  equation  between 
the  average  kinetic  energy,  the  modulus,  and  the  average  index 
of  probability  of  velocity,  taken  negatively,  is  identical  with 
that  given  by  Clausius  locis  citatis  for  the  kinetic  energy  of  a 
body,  the  temperature,  and  a  quantity  which  he  called  the 
transformation-value  of  the  kinetic  energy,  f  The  relations 


*  Pogg.  Ann.,  Bd.  CXVI,  S.  73,  (1862)  ;  ibid.,  Bd.  CXXV,  S.  353,  (1865), 
See  also  Boltzmann,  Sitzb.  der  Wiener.Akad.,  Bd.  LXIII,  S.  728,  (1871). 
t  Verwandlungswerth  des  Warmeinhaltes. 


70  AVERAGE   VALUES  IN  A    CANONICAL 

are  also  identical  with  those  given  by  Clausius  for  the  corre- 
sponding quantities. 

Equations  (112)  and  (181)  show  that  if  ty  or  ^rq  is  known 
as  function  of  S  and  «x  ,  a2  ,  etc.,  we  can  obtain  by  differentia- 
tion e  or  eq,  and  Aly  AZy  etc.  as  functions  of  the  same  varia- 
bles. We  have  in  fact 


*  =  *f-«i=:*f-e.  (192) 

The  corresponding  equation  relating  to  kinetic  energy, 


which  may  be  obtained  in  the  same  way,  may  be  verified  by 
the  known  relations  (186),  (187),  and  (188)  between  the 
variables.  We  have  also 


etc.,  so  that  the  average  values  of  the  external  forces  may  be 
derived  alike  from  ty  or  from  tyq. 

The  average  values  of  the  squares  or  higher  powers  of  the 
energies  (total,  potential,  or  kinetic)  may  easily  be  obtained  by 
repeated  differentiations  of  -\|r,  ^,  ^p1  or  e,  eg,  e^,  with 
respect  to  <t).  By  equation  (108)  we  have 


c  =  J  . .  .J  «  e  <fe .  . .  dfc,  (195) 

phases 

and  differentiating  with  respect  to  ®, 


phases 

whence,  again  by  (108), 

de  _  ?  —  \fe 
d®~~      ®2 


ENSEMBLE   OF  SYSTEMS.  71 


= 

Combining  this  with  (191), 


In  precisely  the  same  way,  from  the  equation 

, 
^...^n,  (199) 


(200) 


config. 

we  may  obtain 


In  the  same  way  also,  if  we  confine  ourselves  to  a  particular 
configuration,  from  the  equation 


/.all     r      ^       1 

=  /•••/  €Pe        Ap  dpi . .  .  dpM  (201) 

J       J 

we  obtain 


€r     J 

veloc. 


which  by  (187)  reduces  to 

?=(!n»+Jn)®».  (203) 

Since  this  value  is  independent  of  the  configuration,  we  see 
that  the  average  square  of  the  kinetic  energy  for  every  configu- 
ration is  the  same,  and  therefore  the  same  as  for  the  whole 
ensemble.  Hence  e^  may  be  interpreted  as  the  average  either 
for  any  particular  configuration,  or  for  the  whole  ensemble. 
It  will  be  observed  that  the  value  of  this  quantity  is  deter- 
mined entirely  by  the  modulus  and  the  number  of  degrees  of 
freedom  of  the  system,  and  is  in  other  respects  independent  of 
the  nature  of  the  system. 

Of  especial  importance  are  the  anomalies  of  the  energies,  or 
their  deviations  from  their  average  values.    The  average  value 


72  AVERAGE   VALUES  IN  A    CANONICAL 

of  these  anomalies  is  of  course  zero.     The  natural  measure  of 
such  anomalies  is  the  square  root  of  their  average  square.    Now 


(.-•3"  =  ?_,  (204) 

identically.    Accordingly 


(205) 
In  like  manner, 

(206) 


Hence 

G-l)2  =  Gfl  -  I,)2  +  (ep-ep)2.  (208) 

Equation  (206)  shows  that  the  value  of  deg/d®  can  never  be 
negative,  and  that  the  value  of  d2tyg/d®2  or  drjq/d®  can  never 
be  positive.* 

To  get  an  idea  of  the  order  of  magnitude  of  these  quantities, 
we  may  use  the  average  kinetic  energy  as  a  term  of  comparison, 
this  quantity  being  independent  of  the  arbitrary  constant  in- 
volved in  the  definition  of  the  potential  energy.  Since 

*  In  the  case  discussed  in  the  note  on  page  54,  in  which  the  potential 
energy  is  a  quadratic  function  of  the  q's,  and  Ag  independent  of  the  <?'s,  we 
should  get  for  the  potential  energy 


and  for  the  total  energy 


We  may  also  write  in  this  case, 

(fq  —  «a)2      n 
(e-e0)2~n' 


ENSEMBLE  OF  SYSTEMS.  73 


(209) 


e-__  ?. 

~?~  "»^~»  +  »5fp 

These  equations  show  that  when  the  number  of  degrees  of 
freedom  of  the  systems  is  very  great,  the  mean  squares  of  the 
anomalies  of  the  energies  (total,  potential,  and  kinetic)  are  very 
small  in  comparison  with  the  mean  square  of  the  kinetic 
energy,  unless  indeed  the  differential  coefficient  deq/dep  is 
of  the  same  order  of  magnitude  as  n.  Such  values  of  deqjdep 
can  only  occur  within  intervals  (ej1  —  epf)  which  are  of  the  or- 
der of  magnitude  of  n~~\  unless  it  be  in  cases  in  which  eg  is  in 
general  of  an  order  of  magnitude  higher  than  ep.  Postponing 
for  the  moment  the  consideration  of  such  cases,  it  will  be  in- 
teresting to  examine  more  closely  the  case  of  large  values  of 
deq/dep  within  narrow  limits.  Let  us  suppose  that  for  ej  and 
epf  the  value  of  deq/dep  is  of  the  order  of  magnitude  of  unity, 
but  between  these  values  of  "ep  very  great  values  of  the  differ- 
ential coefficient  occur.  Then  in  the  ensemble  having  modulus 
@"  and  average  energies  ep"  and  es",  values  of  eq  sensibly  greater 
than  eqrl  will  be  so  rare  that  we  may  call  them  practically  neg- 
ligible. They  will  be  still  more  rare  in  an  ensemble  of  less 
modulus.  For  if  we  differentiate  the  equation 


regarding  eq  as  constant,  but  ®  and  therefore  ^  as  variable, 
we  get 

/drjq\    __1  dif/q       \Itq  —  €q  . 

\d®)€-®~d®         ©^~' 
whence  by  (192) 


74  AVERAGE   VALUES  IN  A   CANONICAL 

That  is,  a  diminution  of  the  modulus  will  diminish  the  proba- 
bility of  all  configurations  for  which  the  potential  energy  exceeds 
its  average  value  in  the  ensemble.  Again,  in  the  ensemble 
having  modulus  ®'  and  average  energies  ep'  and  e^,  values  of 
eq  sensibly  less  than  eg'  will  be  so  rare  as  to  be  practically  neg- 
ligible. They  will  be  still  more  rare  in  an  ensemble  of  greater 
modulus,  since  by  the  same  equation  an  increase  of  the 
modulus  will  diminish  the  probability  of  configurations  for 
which  the  potential  energy  is  less  than  its  average  value  in 
the  ensemble.  Therefore,  for  values  of  O  between  ®'  and  ®", 
and  of  ep  between  ep'  and  ep/;,  the  individual  values  of  eq  will 
be  practically  limited  to  the  interval  between  e«/  and  eg'r. 

In  the  cases  which  remain  to  be  considered,  viz.,  when 
deq/dep  has  very  large  values  not  confined  to  narrow  limits, 
and  consequently  the  differences  of  the  mean  potential  ener- 
gies in  ensembles  of  different  moduli  are  in  general  very  large 
compared  with  the  differences  of  the  mean  kinetic  energies,  it 
appears  by  (210)  that  the  anomalies  of  mean  square  of  poten- 
tial energy,  if  not  small  in  comparison  with  the  mean  kinetic 
energy,  will  yet  in  general  be  very  small  in  comparison  with 
differences  of  mean  potential  energy  in  ensembles  having 
moderate  differences  of  mean  kinetic  energy,  —  the  exceptions 
being  of  the  same  character  as  described  for  the  case  when 
deq/dep  is  not  in  general  large. 

It  follows  that  to  human  experience  and  observation  with 
respect  to  such  an  ensemble  as  we  are  considering,  or  with 
respect  to  systems  which  may  be  regarded  as  taken  at  random 
from  such  an  ensemble,  when  the  number  of  degrees  of  free- 
dom is  of  such  order  of  magnitude  as  the  number  of  molecules 
in  the  bodies  subject  to  our  observation  and  experiment,  e  —  e, 
€P  —  £pi  *q  —  %  would  be  in  general  vanishing  quantities, 
since  such  experience  would  not  be  wide  enough  to  embrace 
the  more  considerable  divergencies  from  the  mean  values,  and 
such  observation  not  nice  enough  to  distinguish  the  ordinary 
divergencies.  In  other  words,  such  ensembles  would  appear 
to  human  observation  as  ensembles  of  systems  of  uniform 
energy,  and  in  which  the  potential  and  kinetic  energies  (sup- 


ENSEMBLE   OF  SYSTEMS.  75 

posing  that  there  were  means  of  measuring  these  quantities 
separately)  had  each  separately  uniform  values.*  Exceptions 
might  occur  when  for  particular  values  of  the  modulus  the 
differential  coefficient  deq/d~ep  takes  a  very  large  value.  To 
human  observation  the  effect  would  be,  that  in  ensembles  in 
which  ®  and  ep  had  certain  critical  values,  ~eq  would  be  in- 
determinate within  certain  limits,  viz.,  the  values  which  would 
correspond  to  values  of  ®  and  ep  slightly  less  and  slightly 
greater  than  the  critical  values.  Such  indeterminateness  cor- 
responds precisely  to  what  we  observe  in  experiments  on  the 
bodies  which  nature  presents  to  us.f 

To  obtain  general  formulae  for  the  average  values  of  powers 
of  the  energies,  we  may  proceed  as  follows.  If  h  is  any  posi- 
tive whole  number,  we  have  identically 


phases  phases 

t.  e.,  by  (108), 

_i        ,,         _i 

(215) 


Hence 


and 


*  This  implies  that  the  kinetic  and  potential  energies  of  individual  systems 
would  each  separately  have  values  sensibly  constant  in  time. 

t  As  an  example,  we  may  take  a  system  consisting  of  a  fluid  in  a  cylinder 
under  a  weighted  piston,  with  a  vacuum  between  the  piston  and  the  top  of 
the  cylinder,  which  is  closed.  The  weighted  piston  is  to  be  regarded  as  a 
part  of  the  system.  (This  is  formally  necessary  in  order  to  satisfy  the  con- 
dition of  the  invariability  of  the  external  coordinates.)  It  is  evident  that  at 
a  certain  temperature,  viz.,  when  the  pressure  of  saturated  vapor  balances 
the  weight  of  the  piston,  there  is  an  indeterminateness  in  the  values  of  the 
potential  and  total  energies  as  functions  of  the  temperature. 


76  AVERAGE    VALUES  IN  A    CANONICAL 

For  h  =  1,  this  gives 


which  agrees  with  (191). 
From  (215)  we  have  also 


In  like  manner  from  the  identical  equation 

all    ,  «, 


config.  conflg. 

(221) 

--/      rf\^  -— 
we  get  i?  =  e  0  (^©2  ^  J  e     ® ,  (222) 


and 


With  respect  to  the  kinetic  energy  similar  equations  will 
hold  for  averages  taken  for  any  particular  configuration,  or 
for  the  whole  ensemble.  But  since 


the  equation 


reduces  to 


ENSEMBLE   OF  SYSTEMS.  77 

We  have  therefore 

<226> 
"          <227> 

*(228) 


The  average  values  of  the  powers  of  the  anomalies  of  the 
energies  are  perhaps  most  easily  found  as  follows.  We  have 
identically,  since  e  is  a  function  of  ®,  while  e  is  a  function  of 
the  jt?'s  and  <?'s, 

all  f 


phases 


J.  .  .  J[e(e  _  i)»  _  h  (e  _  ;)«  ®2*  J  e~0dPl,  ...dy. 


(229) 
_    i_   x  enyj 

phases 

i.  e.,  by  (108), 


•  (230) 

*  In  the  case  discussed  in  the  note  on  page  54  we  may  easily  get 


which,  with  eg  —  60  —  „  ®, 

gives 

rr^j»  =  Qe  +  e»^)  («,-*j«  =  |Qe  +  «         * 

Hence  c  —  eaft  =  c*. 


Again  (e  -  60)»  =     e  -  ea  +  02^     (e  -  ea)*-1, 

which  with  e  —  e0  =  n  & 

gives 

(e  -  ««)*  =  (n  6  +  02^)  (e  -  ea)*-1  =  n  (w  0  +  02^)*~J0, 

hence  {7^j»  =  ?^  +  *>  e». 


78  AVERAGE   VALUES  IN  A   CANONICAL 

or  since  by  (218) 


-e)»«  =  e(e-e)»  -  A  <«- 


In  precisely  the  same  way  we  may  obtain  for  the  potential 
energy 

(63-i3)^  =  @2^(e3-  eq^  +  h(eq-  eq)^  ©2g.     (232) 
By  successive  applications  of  (231)  we  obtain 


(e  -  i)2  = 
(e-e)8  =• 


(e  -  e)6  =  J>5e  +  15DeD*e  +  10(D2€)2  +  15(Z)e)8  etc. 

where  D  represents  the  operator  ®'2d/d®.  Similar  expres- 
sions relating  to  the  potential  energy  may  be  derived  from 
(232). 

For  the  kinetic  energy  we  may  write  similar  equations  in 
which  the  averages  may  be  taken  either  for  a  single  configura- 
tion or  for  the  whole  ensemble.  But  since 

d€p  _  n 

d®~2 

the  general  formula  reduces  to 

(ep  -  ep)™  =  ©2  A  (€p  -  ep)»  +  ±nh&  (ep  - ~ep)^    (233) 
or 


(234) 


ENSEMBLE  OF  SYSTEMS.  79 

But  since  identically 


the  value  of  the  corresponding  expression  for  any  index  will 
be  independent  of  <*)  and  the  formula  reduces  to 


we  have  therefore 


etc.1 


It  will  be  observed  that  when  i/r  or  e  is  given  as  function  of 
O,  all  averages  of  the  form  e^  or  (e  —  T)ft  are  thereby  deter- 

*  In  the  case  discussed  in  the  preceding  foot-notes  we  get  easily 


and 


For  the  total  energy  we  have  in  this  case 

l      h  ~ 


x±-Tx2      i 
Ve-J    =n' 


ft  —  €\  °  _   2 


etc. 


rurxs  iar  A 


•  ou 


-:     .• 


/ 

f. 

J 


*    «»» 


ENSEMBLE  OP  SYSTEMS. 


The  multiple  integrals  in 

average  rallies  of  the  expressions  In  the  brackets, 

may  therefore  set  equal  to  zero.    The  first  gives 


as  already  obtained.    With  this  relation  and  (191)  we  get 
from  the  other  equations 


We  may  add  for  comparison  equation  (205),  which  might  be 
derived  from  (236)  by  differentiating  twice  with  respect  to  8  : 


The  two  last  equations  give 


dl 


(Al  -  Al)(e  -  e)  =  —  (6  -  €)'.  (245) 

e?e 

If  i/r  or  e  is  known  as  function  of  0,  Oj,  Oj,  etc*,  (e  —  e)2  may 
be  obtained  by  differentiation  as  function  of  the  same  variables. 
And  if  i|r,  or  Av  or  17"  is  known  as  function  of  8,  O 


(e  —  e)  may  be  obtained  by  differentiation.     But 
(^Al  —  A^y-  and  (^Al  —  A^)  (^2  —  A2)  cannot  be  obtained  in  any 

similar  manner.  We  have  seen  that  (e—  e)2  is  in  general  a 
vanishing  quantity  for  very  great  values  of  TI,  which  we  may 
regard  as  contained  implicitly  in  0  as  a  divisor.  The  same  is 

true  of  (A^  —  A^)  (e  —  e).     It  does  not  appear  that  we  can 

assert  the  same  of  (A-^  —  -4X)2  or  (Al  —  A^)  (^2  —  -42),  since 

6 


82  AVERAGE   VALUES  IN  A    CANONICAL 


a^  may  be  very  great.  The  quantities  dte/da^  an 
belong  to  the  class  called  elasticities.  The  former  expression 
represents  an  elasticity  measured  under  the  condition  that 
while  &J  is  varied  the  internal  coordinates  ql9  .  .  .  qn  all  remain 
fixed.  The  latter  is  an  elasticity  measured  under  the  condi- 
tion that  when  ax  is  varied  the  ensemble  remains  canonically 
distributed  within  the  same  modulus.  This  corresponds  to 
an  elasticity  in  physics  measured  under  the  condition  of  con- 
stant temperature.  It  is  evident  that  the  former  is  greater 
than  the  latter,  and  it  may  be  enormously  greater. 

The  divergences  of  the  force  Al  from  its  average  value  are 
due  in  part  to  the  differences  of  energy  in  the  systems  of  the 
ensemble,  and  in  part  to  the  differences  in  the  value  of 
the  forces  which  exist  in  systems  of  the  same  energy.  If  we 
write  A^  for  the  average  value  of  Al  in  systems  of  the 
ensemble  which  have  any  same  energy,  it  will  be  determined 
by  the  equation 


/  .  .  .  J  e  ® 


.  .  .  dqn 


where  the  limits  of  integration  in  both  multiple  integrals  are 
two  values  of  the  energy  which  differ  infinitely  little,  say  e  and 

fc± 

e  +  de.  This  will  make  the  factor  e  &  constant  within  the 
limits  of  integration,  and  it  may  be  cancelled  in  the  numera- 
tor and  denominator,  leaving 

/•••/-  -£-<&>!  ...dqn 

2H.=         /     /  (247) 

J...J*!...*. 

where  the  integrals  as  before  are  to  be  taken  between  e  and 
e  +  de.  A^\f  is  therefore  independent  of  ®,  being  a  function 
of  the  energy  and  the  external  coordinates. 


ENSEMBLE  OF  SYSTEMS.  83 

Now  we  have  identically 

Al  —  Ai  =  (Ai  —  2T)e)  +  (2T1 1  —  -4)> 

where  Al  —  ~A^e  denotes  the  excess  of  the  force  (tending  to 
increase  a^  exerted  by  any  system  above  the  average  of  such 
forces  for  systems  of  the  same  energy.  Accordingly, 


But  the  average  value  of  (Al  —  A^\f)  (A^\ e  —  A^)  for  systems 
of  the  ensemble  which  have  the  same  energy  is  zero,  since  for 
such  systems  the  second  factor  is  constant.  Therefore  the 
average  for  the  whole  ensemble  is  zero,  and 


Atf.  (248) 

In  the  same  way  it  may  be  shown  that 


(A,  -  Al)  (e-e)  =  (^  -  AJ  (e  -  e).  (249) 

It  is  evident  that  in  ensembles  in  which  the  anomalies  of 
energy  e  —  e  may  be  regarded  as  insensible  the  same  will  be 
true  of  the  quantities  represented  by  A^\f  —  A^ 

The  properties  of  quantities  of  the  form  A^\€  will  be 
farther  considered  in  Chapter  X,  which  will  be  devoted  to 
ensembles  of  constant  energy. 

It  may  not  be  without  interest  to  consider  some  general 
formulae  relating  to  averages  in  a  canonical  ensemble,  which 
embrace  many  of  the  results  which  have  been  given  in  this 
chapter. 

Let  u  be  any  function  of  the  internal  and  external  coordi- 
nates with  the  momenta  and  modulus.  We  have  by  definition 

**-.>,V:.fc! 

u-J...Juee  d^.^dq,  (250) 

phases 

If  we  differentiate  with  respect  to  ®,  we  have 
du       f  a    r/du       u  u  e 

d®=J   J  (35-3 <#--^i 

phases 


84  AVERAGE    VALUES  IN  A  CANONICAL 


du  _du       uty-e)       udif, 
d®~d®  --  &—  +  ®d®' 

Setting  u  =  1  in  this  equation,  we  get 

d\f/  _  \i/  —  € 
d®~      0 

and  substituting  this  value,  we  have 

du      du      ue      ue 


If  we  differentiate  equation  (250)  with  respect  to  a  (which 
may  represent  any  of  the  external  coordinates),  and  write  A 

for  the  force  —  -^  ,  we  get 


__  ail  t  *. 

du  r  r(  du  u  dif/  u  .  \ 
3-=  /.../V-5-  +  ^^-+7v^) 
da  J  J  \da  ©  da  0  / 


da 

phases 

du      du 
or  —  =  — 


Setting  w  =  1  hi  this  equation,  we  get 


Substituting  this  value,  we  have 

du      au      uA      uA 


du          du 


or  ®-r-®-r  =  ^2-uI=(u-u)(A-2).        (255) 

da          aa 

Repeated  applications  of  the  principles  expressed  by  equa- 
tions (252)  and  (255)  are  perhaps  best  made  in  the  particular 
cases.  Yet  we  may  write  (252)  in  this  form 


ENSEMBLE  OF  SYSTEMS.  85 


(€  +  D)  (u  -  u)  =  0,  (256) 


where  D  represents  the  operator  ®2  d/d®. 
Hence 


(e  +  D)A  (u  -  u)  =  0,  (257) 

where  h  is  any  positive  whole  number.  It  will  be  observed, 
that  since  e  is  not  function  of  ®,  (e  +  D)h  may  be  expanded  by 
the  binomial  theorem.  Or,  we  may  write 


(e  +  />)  u  =  (e  +  D)  u,  (258) 


whence  (e  +  X>)*  u  =  (e  +  D)h  u.  (259) 

But  the  operator  (e  +  D)*,  although  in  some  respects  more 
simple  than  the  operator  without  the  average  sign  on  the  e, 
cannot  be  expanded  by  the  binomial  theorem,  since  e  is  a 
function  of  ®  with  the  external  coordinates. 
So  from  equation  (254)  we  have 


<26°) 


whence  (~  +  J;)*  («  -  u)  =  0  ;  (261) 


The  binomial  theorem  cannot  be  applied  to  these  operators. 

Again,  if  we  now  distinguish,  as  usual,  the  several  external 
coordinates  by  suffixes,  we  may  apply  successively  to  the 
expression  u  —  u  any  or  all  of  the  operators 


, 


,    etc.       (264) 


86  AVERAGES  IN  A   CANONICAL  ENSEMBLE. 

as  many  times  as  we  choose,  and  in  any  order,  the  average 
value  of  the  result  will  be  zero.  Or,  if  we  apply  the  same 
operators  to  u,  and  finally  take  the  average  value,  it  will  be  the 
same  as  the  value  obtained  by  writing  the  sign  of  average 
separately  as  u,  and  on  e,  A± ,  A2 ,  etc.,  in  all  the  operators. 

If  u  is  independent  of  the  momenta,  formulae  similar  to 
the  preceding,  but  having  eq  in  place  of  e,  may  be  derived 
from  equation  (179). 


CHAPTER  VIII. 

ON  CERTAIN  IMPORTANT  FUNCTIONS  OF  THE 
ENERGIES  OF  A  SYSTEM. 

IN  order  to  consider  more  particularly  the  distribution  of  a 
canonical  ensemble  in  energy,  and  for  other  purposes,  it  will 
be  convenient  to  use  the  following  definitions  and  notations. 

Let  us  denote  by  J^the  extension-in-phase  below  a  certain 
limit  of  energy  which  we  shall  call  e.     That  is,  let 

>x . .  .  dqn,  (265) 

the  integration  being  extended  (with  constant  values  of  the 
external  coordinates)  over  all  phases  for  which  the  energy  is 
less  than  the  limit  e.  We  shall  suppose  that  the  value  of  this 
integral  is  not  infinite,  except  for  an  infinite  value  of  the  lim- 
iting energy.  This  will  not  exclude  any  kind  of  system  to 
which  the  canonical  distribution  is  applicable.  For  if 

>i  •  •  •  dqn 

taken  without  limits  has  a  finite  value,*  the  less  value  repre- 
sented by 

e 


/... 

u 


• 


taken  below  a  limiting  value  of  6,  and  with  the  e  before  the 
integral  sign  representing  that  limiting  value,  will  also  be 
finite.  Therefore  the  value  of  V,  which  differs  only  by  a 
constant  factor,  will  also  be  finite,  for  finite  e.  It  is  a  func- 
tion of  e  and  the  external  coordinates,  a  continuous  increasing 

*  This    is  a  necessary  condition    of    the  canonical    distribution.      See 
Chapter  IV,  p.  35. 


88  CERTAIN  IMPORTANT  FUNCTIONS 

function  of  6,  which  becomes   infinite  with  e,  and  vanishes 
for  the  smallest  possible  value  of  e,  or  f or  e  =  —  oo,  if  the 
energy  may  be  diminished  without  limit. 
Let  us  also  set 

dV 
<f>  =  log  —  •  (266) 

The  extension  in  phase  between  any  two  limits  of  energy,  ^ 
and  e",  will  be  represented  by  the  integral 

/  de.  (267) 

And  in  general,  we  may  substitute  e*  de  for  dpl . . .  dqn  in  a 
2tt-fold  integral,  reducing  it  to  a  simple  integral,  whenever 
the  limits  can  be  expressed  by  the  energy  alone,  and  the  other 
factor  under  the  integral  sign  is  a  function  of  the  energy  alone, 
or  with  quantities  which  are  constant  in  the  integration. 

In  particular  we  observe  that  the  probability  that  the  energy 
of  an  unspecified  system  of  a  canonical  ensemble  lies  between 
the  limits  e'  and  e"  will  be  represented  by  the  integral  * 

*  0ffe,  (268) 

and  that  the  average  value  in  the  ensemble  of  any  quantity 
which  only  varies  with  the  energy  is  given  by  the  equation  j 


(269) 


where  we  may  regard  the  constant  *fy  as  determined  by  the 
equation  $ 


^» 
=l 


6=00 

— 
& 


e  de,  (270) 

F=0 

In  regard  to  the  lower  limit  in  these  integrals,  it  will  be  ob- 
served that  V=  0  is  equivalent  to  the  condition  that  the 
value  of  e  is  the  least  possible. 

*  Compare  equation  (93).  t  Compare  equation  (108). 

J  Compare  equation  (92). 


OF  THE  ENERGIES   OF  A   SYSTEM.  89 

In  like  manner,  let  us  denote  by  Vq  the  extension-in-configu- 
ration  below  a  certain  limit  of  potential  energy  which  we  may 
call  eg.  That  is,  let 


•  JV 


(2T1) 


the  integration  being  extended  (with  constant  values  of  the 
external  coordinates)  over  all  configurations  for  which  the 
potential  energy  is  less  than  eg.  Vq  will  be  a  function  of  eq 
with  the  external  coordinates,  an  increasing  function  of  e3, 
which  does  not  become  infinite  (in  such  cases  as  we  shall  con- 
sider *)  for  any  finite  value  of  eq.  It  vanishes  for  the  least 
possible  value  of  e?,  or  for  eq  =  —  oo  ,  if  eq  can  be  diminished 
without  limit.  It  is  not  always  a  continuous  function  of  eg. 
In  fact,  if  there  is  a  finite  extension-in-configuration  of  con- 
stant potential  energy,  the  corresponding  value  of  Vq  will 
not  include  that  extension-in-configuration,  but  if  eq  be  in- 
creased infinitesimally,  the  corresponding  value  of  Vq  will  be 
increased  by  that  finite  extension-in-configuration. 
Let  us  also  set 


(272) 

The  extension-in-configuration  between  any  two  limits  of 
potential  energy  eq  and  eqf  may  be  represented  by  the  integral 

(273) 

whenever  there  is  no  discontinuity  in  the  value  of  Vq  as 
function  of  eq  between  or  at  those  limits,  that  is,  when- 
ever there  is  no  finite  extension-in-configuration  of  constant 
potential  energy  between  or  at  the  limits.  And  hi  general, 
with  the  restriction  mentioned,  we  may  substitute  e^q  deq  for 
Aj  dq1  .  .  .  dqn  in  an  w-fold  integral,  reducing  it  to  a  simple 
integral,  when  the  limits  are  expressed  by  the  potential  energy, 
and  the  other  factor  under  the  integral  sign  is  a  function  of 

*  If  Vq  were  infinite^  for  finite  values  of  e,,  V  would  evidently  be  infinite 
for  finite  values  of  e. 


90  CERTAIN  IMPORTANT  FUNCTIONS 

the  potential  energy,  either  alone  or  with  quantities  which  are 
constant  in  the  integration. 

We  may  often  avoid  the  inconvenience  occasioned  by  for- 
mulae becoming  illusory  on  account  of  discontinuities  in  the 
values  of  Vq  as  function  of  eq  by  substituting  for  the  given 
discontinuous  function  a  continuous  function  which  is  practi- 
cally equivalent  to  the  given  function  for  the  purposes  of  the 
evaluations  desired.  It  only  requires  infinitesimal  changes  of 
potential  energy  to  destroy  the  finite  extensions-in-configura- 
tion  of  constant  potential  energy  which  are  the  cause  of  the 
difficulty. 

In  the  case  of  an  ensemble  of  systems  canonically  distributed 
in  configuration,  when  Vq  is,  or  may  be  regarded  as,  a  continu- 
ous function  of  eq  (within  the  limits  considered),  the  proba- 
bility that  the  potential  energy  of  an  unspecified  system  lies 
between  the  limits  eq  and  eq'  is  given  by  the  integral 


where  ^  may  be  determined  by  the  condition  that  the  value  of 
the  integral  is  unity,  when  the  limits  include  all  possible 
values  of  eq.  In  the  same  case,  the  average  value  in  the  en- 
semble of  any  function  of  the  potential  energy  is  given  by  the 
equation 


u  =  /  ue  d€q.  (275) 

Vq=0 

When  Vq  is  not  a  continuous  function  of  eff,  we  may  write  d  Vq 
for  e*qdeg  in  these  formulae. 

In  like  manner  also,  for  any  given  configuration,  let  us 
denote  by  Vp  the  extension-in-velocity  below  a  certain  limit  of 
kinetic  energy  specified  by  ep.  That  is,  let 


V,  =  J. 


(276) 


OF  THE  ENERGIES  OF  A   SYSTEM.  91 

the  integration  being  extended,  with  constant  values  of  the 
coordinates,  both  internal  and  external,  over  all  values  of  the 
momenta  for  which  the  kinetic  energy  is  less  than  the  limit  ep. 
Vp  will  evidently  be  a  continuous  increasing  function  of  ep 
which  vanishes  and  becomes  infinite  with  e.  Let  us  set 


The  extension-in-velocity  between  any  two  limits  of  kinetic 
energy  ep  and  ep"  may  be  represented  by  the  integral 

f 

e*pdep.  (278) 

And  in  general,  we  may  substitute  e^p  dep  for  A,*  dpl  .  .  .  dpn 
or  Ag*  dql  .  .  .  dqn  in  an  w-fold  integral  in  which  the  coordi- 
nates are  constant,  reducing  it  to  a  simple  integral,  when  the 
limits  are  expressed  by  the  kinetic  energy,  and  the  other  factor 
under  the  integral  sign  is  a  function  of  the  kinetic  energy, 
either  alone  or  with  quantities  which  are  constant  in  the 
integration. 

It  is  easy  to  express  Vp  and  $p  in  terms  of  ep.     Since  A^  is 
function  of  the  coordinates  alone,  we  have  by  definition 


1...dpn  (279) 

the  limits  of  the  integral  being  given  by  ep.     That  is,  if 

ep  =  F(Pl,...Pa),  (280) 

the  limits  of  the  integral  for  ep  =  1,  are  given  by  the  equation 
F(Pl,...Pa)  =  \,  (281) 

and  the  limits  of  the  integral  for  ep  —  a2,  are  given  by  the 
equation 

=«'.  (282) 


But  since  F  represents  a  quadratic  function,  this  equation 
may  be  written 

1  (283) 


92  CERTAIN  IMPORTANT  FUNCTIONS 

The  value  of  Vp  may  also  be  put  in  the  form 

r,  =  ***f...f*&...*%.  (284) 

Now  we  may  determine  Vp  for  ep  =  1  from  (279)  where  the 
limits  are  expressed  by  (281),  and  FJ,  for  ep  ,=  a2  from  (284) 
taking  the  limits  from  (283).  The  two  integrals  thus  deter- 
mined are  evidently  identical,  and  we  have 


(285) 

i.  e.,  Vv  varies  as  e/.     We  may  therefore  set 

,  n 

Vp=Cep*>  eP  =  n-Cep*      j  (286) 

where  C  is  a  constant,  at  least  for  fixed  values  of  the  internal 
coordinates. 

To  determine  this  constant,  let  us  consider  the  case  of  a 
canonical  distribution,  for  which  we  have 


_ 

where  e&  =  (2-*®)  2. 

Substituting  this  value,  and  that  of  e*'  from  (286),  we  get 


(287) 


Having  thus  determined  the  value  of  the  constant  (7,  we  may 


OF  THE  ENERGIES  OF  A   SYSTEM.  -     93 

substitute  it  in  the  general  expressions  (286),  and  obtain  the 
following  values,  which  are  perfectly  general  : 


~  *(289) 

It  will  be  observed  that  the  values  of  Vp  and  <f>p  for  any 
given  ep  are  independent  of  the  configuration,  and  even  of  the 
nature  of  the  system  considered,  except  with  respect  to  its 
number  of  degrees  of  freedom. 

Returning  to  the  canonical  ensemble,  we  may  express  the 
probability  that  the  kinetic  energy  of  a  system  of  a  given 
configuration,  but  otherwise  unspecified,  falls  within  given 
limits,  by  either  member  of  the  following  equation 


Since  this  value  is  independent  of  the  coordinates  it  also 
represents  the  probability  that  the  kinetic  energy  of  an 
unspecified  system  of  a  canonical  ensemble  falls  within  the 
limits.  The  form  of  the  last  integral  also  shows  that  the  prob- 
ability that  the  ratio  of  the  kinetic  energy  to  the  modulus 

*  Very  similar  values  for  Vq,  <&*,  V,  and  e*  may  be  found  in  the  same 
way  in  the  case  discussed  in  the  preceding  foot-notes  (see  pages  54, 72,  77,  and 
79),  in  which  e3  is  a  quadratic  function  of  the  q's,  and  Aj  independent  of  the  q'a. 
In  this  case  we  have 


(2  ')*(««  - 


P(Jn) 


+  i) 


94  CERTAIN  IMPORTANT  FUNCTIONS 

falls  within  given  limits  is  independent  also  of  the  value  of 
the  modulus,  being  determined  entirely  by  the  number  of 
degrees  of  freedom  of  the  system  and  the  limiting  values 
of  the  ratio. 

The  average  value  of  any  function  of  the  kinetic  energy, 
either  for  the  whole  ensemble,  or  for  any  particular  configura- 
tion, is  given  by 

€p 

—•£    ?-i 
ue    0e,2      dep  *(291) 


Thus: 

^®"'      if      m  +  ^>°>      t(292) 


*  The  corresponding  equation  for  the  average  value  of  any  function  of 
the  potential  energy,  when  this  is  a  quadratic  function  of  the  ^'s,  and  A£  is 
independent  of  the  q's,  is 


In  the  same  case,  the  average  value  of  any  function  of  the  (total)  energy  is 
given  by  the  equation 


Hence  in  this  case 


j      .f      m  +  n>0- 


and  =     ,        if 


ii  f         vy 

If  n  =  1,  e*  =  2  ir   and   d^jde  =  0  for  any  value  of  e.    If  n  =  2,  the  case  is 
the  same  with  respect  to  02. 

t  This  equation  has  already  been  proved  for  positive  integral  powers  of 
the  kinetic  energy.    See  page  77. 


OF  THE  ENERGIES  OF  A   SYSTEM.  95 


n     n 


-)  /o  _\9  ^2    ~j  if          w  >  1  ;  (294) 

if        n  >  2 ;  (295) 

=  ©.  (296) 

If  n  =  2,  e*p  =  2  TT,  and  d<j>p/dep  =  0,  for  any  value  of  ep. 
The  definitions  of  F,  V#  and  F^,  give 


(297) 

where  the  integrations  cover  all  phases  for  which  the  energy 
is  less  than  the  limit  e,  for  which  the  value  of  Fis  sought. 
This  gives 


V=CvpdVq,  (298) 

and  ,-j-r     €9=6 

e*  =  -~  —  f  e^pdVn,  (299} 

de      j 

where  Vp  and  e^p  are  connected  with  Vq  by  the  equation 

€p  +  eq  =  constant  ~  e.  (300) 

If  n  >  2,  e*?  vanishes  at  the  upper  limit,  i.  e.,  for  ep  =  0,  and 
we  get  by  another  differentiation 


€q=€ 


We  may  also  write 

62= e 

F=  J     "P;/9^,  (302) 


*    r 
°=J 


(303) 


96  CERTAIN  IMPORTANT  FUNCTIONS 

etc.,  when  Vq  is  a  continuous  function  of  eq  commencing  with 
the  value  Vq  =  0,  or  when  we  choose  to  attribute  to  Vq  a 
fictitious  continuity  commencing  with  the  value  zero,  as  de- 
scribed on  page  90. 

If  we  substitute  hi  these  equations  the  values  of  Vp  and  e^p 
which  we  have  found,  we  get 


^=  r/il   /^     <«  -  <«)   <*  ^« '  (304) 


(305) 


where  e^«  c?eg  may  be  substituted  for  d  Vq  in  the  cases  above 
described.  If,  therefore,  n  is  known,  and  Vq  as  function  of 
€p  V  and  e^  may  be  found  by  quadratures. 

It  appears  from  these  equations  that  F"is  always  a  continu- 
ous increasing  function  of  e,  commencing  with  the  value  V= 
0,  even  when  this  is  not  the  case  with  respect  to  Vq  and  eq. 
The  same  is  true  of  e^,  when  n  >  2,  or  when  n  =  2  if  Vq  in- 
creases continuously  with  eq  from  the  value  Vq  =  0. 

The  last  equation  may  be  derived  from  the  preceding  by 
differentiation  with  respect  to  e.  Successive  differentiations 
give,  if  h  <  }  n  +  1, 


dhVjd<?  is  therefore  positive  if   A  <  J  n  +  1.     It  is  an  in- 
creasing function  of  e,  if  h  <  Jw.     If  e  is   not  capable  of 
being  diminished  without  limit,   dhVjd^  vanishes  for   the 
least  possible  value  of  e,  if  h  <  \n. 
If  n  is  even, 

n 

(307) 


OF  THE  ENERGIES  OF  A   SYSTEM.  97 


That  is,  V  is  the  same  function  of  e#  as  -  —  -  —  —  of  e. 


When  n  is  large,  approximate  formulae  will  be  more  avail- 
able. It  will  be  sufficient  to  indicate  the  method  proposed, 
without  precise  discussion  of  the  limits  of  its  applicability  or 
of  the  degree  of  its  approximation.  For  the  value  of  e^  cor- 
responding to  any  given  e,  we  have 


/  =  e 


*  deq  =     6**+**  dep,  (308) 


where  the  variables  are  connected  by  the  equation  (300). 
The  maximum  value  of  <f)p  +  <f>Q  is  therefore  characterized  by 
the  equation 

(309) 


de,       de, 


The  values  of  ep  and  eq  determined  by  this  maximum  we  shall 
distinguish  by  accents,  and  mark  the  corresponding  values  of 
functions  of  ep  and  eq  in  the  same  way.  Now  we  have  by 
Taylor's  theorem 


If  the  approximation  is  sufficient  without  going  beyond  the 
quadratic  terms,  since  by  (300) 

€P  ~€P'  =  -  (e*  -  «/)» 
we  may  write 

+^(d^P\'(d\}'-\(^ii^ 

2  *.»  (312> 


where  the  limits  have  been  made  ±  oo  for  analytical  simplicity. 
This  is  allowable  when  the  quantity  in  the  square  brackets 
has  a  very  large  negative  value,  since  the  part  of  the  integral 

7 


98  CERTAIN  IMPORTANT  FUNCTIONS 

corresponding  to  other  than  very  small  values  of  eq  —  eqf  may 
be  regarded  as  a  vanishing  quantity. 
This  gives 

>  _  A/+V  /-ON 

(313) 


or 

^V+^'  +  ilog(2,)-ilog[-(^)'-(^)'].     (3U) 

From  this  equation,  with  (289),  (300)  and  (309),  we  .may 
determine  the  value  of  $  corresponding  to  any  given  value  of 
e,  when  <j>q  is  known  as  function  of  eq. 

Any  two  systems  may  be  regarded  as  together  forming  a 
third  system.  If  we  have  F  or  $  given  as  function  of  e  for 
any  two  systems,  we  may  express  by  quadratures  J^and  $  for 
the  system  formed  by  combining  the  two.  If  we  distinguish 
by  the  suffixes  (  )x,  (  )2,  (  )12  the  quantities  relating  to 
the  three  systems,  we  have  easily  from  the  definitions  of  these 
quantities 


=ff 


(sis) 


$12  |   04>*f7T7'   /   p^1  fj  T7"   /    n^1  '   ^2x7  /O-1  £\ 

«/  «/  «y 

where  the  double  integral  is  to  be  taken  within  the  limits 

Vi  =  0,  V2  =  0,  and  el  +  e2  =  e12 , 

and  the  variables  in  the  single  integrals  are  connected  by  the 
last  of  these  equations,  while  the  limits  are  given  by  the  first 
two,  which  characterize  the  least  possible  values  of  e1  and  e2 
respectively. 

It  will  be  observed  that  these  equations  are  identical  in 
form  with  those  by  which  F'and  $  are  derived  from  Vp  or  cf>p 
and  Vq  or  <f>q,  except  that  they  do  not  admit  in  the  general 
case  those  transformations  which  result  from  substituting  for 
Vp  or  (f>p  the  particukr  functions  which  these  symbols  always 
represent. 


OF  THE  ENERGIES  OF  A   SYSTEM.  99 

Similar  formulae  may  be  used  to  derive  Vq  or  <j>q  for  the 
compound  system,  when  one  of  these  quantities  is  known. 
as  function  of  the  potential  energy  in  each  of  the  systems 
combined. 

The  operation  represented  by  such  an  equation  as 


C 

=    I 


01    02 

e    e 


is  identical  with  one  of  the  fundamental  operations  of  the 
theory  of  errors,  viz.,  that  of  finding  the  probability  of  an  error 
from  the  probabilities  of  partial  errors  of  which  it  is  made  up. 
It  admits  a  simple  geometrical  illustration. 

We  may  take  a  horizontal  line  as  an  axis  of  abscissas,  and  lay 
off  61  as  an  abscissa  measured  to  the  right  of  any  origin,  and 
erect  e^i  as  a  corresponding  ordinate,  thus  determining  a  certain 
curve.  Again,  taking  a  different  origin,  we  may  lay  off  e2  as 
abscissas  measured  to  the  left,  and  determine  a  second  curve  by 
erecting  the  ordinates  e^.  We  may  suppose  the  distance  be- 
tween the  origins  to  be  e12,  the  second  origin  being  to  the  right 
if  e12  is  positive.  We  may  determine  a  third  curve  by  erecting 
at  every  point  in  the  line  (between  the  least  values  of  ei  and  e2) 
an  ordinate  which  represents  the  product  of  the  two  ordinates 
belonging  to  the  curves  already  described-  The  area  between 
this  third  curve  and  the  axis  of  abscissas  will  represent  the  value 
of  e^12.  To  get  the  value  of  this  quantity  for  varying  values 
of  612,  we  may  suppose  the  first  two  curves  to  be  rigidly  con- 
structed, and  to  be  capable  of  being  moved  independently.  We 
may  increase  or  diminish  e12  by  moving  one  of  these  curves  to 
the  right  or  left.  The  third  curve  must  be  constructed  anew 
for  each  different  value  of  e12. 


CHAPTER  IX. 
THE  FUNCTION  <£  AND  THE  CANONICAL  DISTRIBUTION. 

IN  this  chapter  we  shall  return  to  the  consideration  of  the 
canonical  distribution,  in  order  to  investigate  those  properties 
which  are  especially  related  to  the  function  of  the  energy 
which  we  have  denoted  by  </>. 

If  we  denote  by  JV,  as  usual,  the  total  number  of  systems 
in  the  ensemble, 


will  represent  the  number  having  energies  between  the  limits 
e  and  e  +  de.     The  expression 


Ne 


(317) 


represents  what  may  be  called  the  density-in-energy.      This 
vanishes  for  e  =  GO,  for  otherwise  the  necessary  equation 


(318) 


could  not  be  fulfilled.  For  the  same  reason  the  density-in- 
energy  will  vanish  for  e  =  —  co,  if  that  is  a  possible  value  of 
the  energy.  Generally,  however,  the  least  possible  value  of 
the  energy  will  be  a  finite  value,  for  which,  if  n  >  2,  e*  will 
vanish,*  and  therefore  the  density-in-energy.  Now  the  density- 
in-energy  is  necessarily  positive,  and  since  it  vanishes  for 
extreme  values  of  the  energy  if  n  >  2,  it  must  have  a  maxi- 
mum in  such  cases,  in  which  the  energy  may  be  said  to  have 

*  See  page  96. 


THE  FUNCTION  0.  101 

its  most  common  or  most  probable  value,  and  which  is 
determined  by  the  equation 

d(f>       1 

de       ©*  ^       ' 

This  value  of  d(f>/de  is  also,  when  n  >  2,  its  average  value 
in  the  ensemble.  For  we  have  identically,  by  integration  by 
parts, 

'''=!+4>r~ 

v'=o     v=o 

If  n  >  2,  the  expression  in  the  brackets,  which  multiplied  by  N 
would  represent  the  density-in-energy,  vanishes  at  the  limits, 
and  we  have  by  (269)  and  (318) 


It  appears,  therefore,  that  for  systems  of  more  tfyan  two  degrees 
of  freedom,  the  average  value  of  d$/de  in  an  eiis^ri^y  canpni-    / 
cally  distributed  is  identical  with  the  value   of  the  same, 
ential  coefficient  as  calculated  for  the  most  .eoavrooi'.  < 
in  the  ensemble,  both  values  being  reciprocals  of  the  modulus. 
Hitherto,  in  our  consideration  of  the  quantities  F",  V#  Vp,  <£, 
</V  4>pi  we  have  regarded  the  external  coordinates  as  constant. 
It  is  evident,  however,  from  their  definitions  that  V  and  <£  are 
in  general  functions  of  the  external  coordinates  and  the  energy 
(e),  that  Vq  and  $g  are  in  general  functions  of  the  external 
coordinates  and  the  potential  energy  (eg).    Vp  and  <f>p  we  have 
found  to  be  functions  of  the  kinetic  energy  (ep)  alone.    In  the 
equation 


-/ 


de,  (322) 


by  which  -vfr  may  be  determined,  O  and  the  external  coordinates 
(contained  implicitly  in  <£)  are  constant  in  the  integration. 
The  equation  shows  that  i|r  is  a  function  of  these  constants. 


102  TH&  FUNCTION  <j>  AND 

If  their  values  are  varied,  we  shall  have  by  differentiation,  if 
n  >2 


v=o 


+  dai  f*4.  e~e+*<le  +  da,  f|*  <f  ®+V  +  etc.     (323)  ' 
J  dci^  J  da2 

V=0  V=Q 

(Since  e*  vanishes  with  F",  when  n  >  2,  there  are  no  terms  due 
to  the  variations  of  the  limits.)     Hence  by  (269) 


or,  since  —  ^  (325) 

© 


<fy  =  ^0  -  ©      -  dox  -  0        da,  -  etc.         (326) 

ttCt^  tt^ 

Comparing  iliis  with  (112),  we  get 


The  first  of  these  equations  might  be  written* 

r)  <328) 


but  must  not  be  confounded  with  the  equation 


d+\  fdf\    (de\ 

^A,«~    W«  W*.. 

which  is  derived  immediately  from  the  identity 

=-\         L\  (330) 


*  See  equations  (321)  and  (104).  Suffixes  are  here  added  to  the  differential 
coefficients,  to  make  the  meaning  perfectly  distinct,  although  the  same  quan- 
tities may  be  written  elsewhere  without  the  suffixes,  when  it  is  believed  that 
there  is  no  danger  of  misapprehension.  The  suffixes  indicate  the  quantities 
which  are  constant  in  the  differentiation,  the  single  letter  a  standing  for  all 
the  letters  a1}  «2,  etc.,  or  all  except  the  one  which  is  explicitly  varied. 


THE   CANONICAL  DISTRIBUTION.  103 

Moreover,  if  we  eliminate  dty  from  (326)  by  the  equation 

d^  =  0^  +  ^d®  +  de,  (331) 

obtained  by  differentiating  (325),  we  get 

de  =  -®dv-®!Jr-dal-®<Q-da2-  etc.,  (332) 

Cia-l  OLa^, 

or  by  (321), 

.      _^  =  ^e  +  ^^  +  ^^  +  etc.  (333) 

de  da,  aa2 

Except  for  the  signs  of  average,  the  second  member  of  this 
equation  is  the  same  as  that  of  the  identity 

«ty  =  ^de  +  ?±dal  +  ^da2  +  etc.  (334) 

de  dal  da2 

For  the  more  precise  comparison  of  these  equations,  we  may 
suppose  that  the  energy  in  the  last  equation  is  some  definite 
and  fairly  representative  energy  in  the  ensemble.  For  this 
purpose  we  might  choose  the  average  energy.  It  will  per- 
haps be  more  convenient  to  choose  the  most  common  energy, 
which  we  shall  denote  by  e0.  The  same  suffix  will  be  applied 
to  functions  of  the  energy  determined  for  this  value.  Our 
identity  then  becomes 


=  de0  +  da,  +  da,  +  etc.      (335) 

\de  J0  \dajo  \da2J0 

It  has  been  shown  that 

?=(^=l,  (336) 

de       \de)0      ©' 

when  n  >  2.  Moreover,  since  the  external  coordinates  have 
constant  values  throughout  the  ensemble,  the  values  of 
d(p/dav  d(f>Jda^  etc.  vary  in  the  ensemble  only  on  account 
of  the  variations  of  the  energy  (e),  which,  as  we  have  seen, 
may  be  regarded  as  sensibly  constant  throughout  the  en- 
semble, when  n  is  very  great.  In  this  case,  therefore,  we  may 
regard  the  average  values 

<25      ~d4 

-5-S    -=-S    etc., 


104  THE  FUNCTION  <£  AND 

as  practically  equivalent  to  the  values  relating  to  the  most 
common  energy 

— —  I  j     (  —  j  j    etc. 
dtti  JQ      \  d&z  J  Q 

In  this  case  also  de  is  practically  equivalent  to  deQ.  We  have 
therefore,  for  very  large  values  of  n, 

—  dri  —  d<f>Q  (337) 

approximately.  That  is,  except  for  an  additive  constant,  —  77 
may  be  regarded  as  practically  equivalent  to  <£0,  when  the 
number  of  degrees  of  freedom  of  the  system  is  very  great. 
It  is  not  meant  by  this  that  the  variable  part  of  rj  +  <£0  is 
numerically  of  a  lower  order  of  magnitude  than  unity.  For 
when  n  is  very  great,  —  77  and  $0  are  very  great,  and  we  can 
only  conclude  that  the  variable  part  of  77  +  <£0  is  insignifi- 
cant compared  with  the  variable  part  of  rj  or  of  <£0,  taken 
separately. 

Now  we  have  already  noticed  a  certain  correspondence 
between  the  quantities  ®  and  77  and  those  which  in  thermo- 
dynamics are  called  temperature  and  entropy.  The  property 
just  demonstrated,  with  those  expressed  by  equation  (336), 
therefore  suggests  that  the  quantities  <f>  and  de/dQ  may  also 
correspond  to  the  thermodynamic  notions  of  entropy  and  tem- 
perature. We  leave  the  discussion  of  this  point  to  a  sub- 
sequent chapter,  and  only  mention  it  here  to  justify  the 
somewhat  detailed  investigation  of  the  relations  of  these 
quantities. 

We  may  get  a  clearer  view  of  the  limiting  form  of  the 
relations  when  the  number  of  degrees  of  freedom  is  indefi- 
nitely increased,  if  we  expand  the  function  <j>  in  a  series 
arranged  according  to  ascending  powers  of  e  —  e0.  This  ex- 
pansion may  be  written 


(f        £) 

(€  ~  ^ 

(338) 


Adding  the  identical  equation 


THE  CANONICAL  DISTRIBUTION.  105 

\/  —  6  ^  —  €Q  6  — 


©  ©  ©         > 


(339) 

Substituting  this  value  in 


which  expresses  the  probability  that  the  energy  of  an  unspeci- 
fied system  of  the  ensemble  lies  between  the  limits  e'  and  e", 
we  get 


-0 

**.          (340) 

When  the  number  of  degrees  of  freedom  is  very  great,  and 
e  —  e0  in  consequence  very  small,  we  may  neglect  the  higher 
powers  and  write* 


i   . 

"  (341) 


This  shows  that  for  a  very  great  number  of  degrees  of 
freedom  the  probability  of  deviations  of  energy  from  the  most 
probable  value  (e0)  approaches  the  form  expressed  by  the 
'law  of  errors.'  With  this  approximate  law,  we  get 

*  If  a  higher  degree  of  accuracy  is  desired  than  is  afforded  by  this  formula, 
it  may  be  multiplied  by  the  series  obtained  from 


by  the  ordinary  formula  for  the  expansion  in  series  of  an  exponential  func- 
tion. There  would  be  no  especial  analytical  difficulty  in  taking  account  of 
a  moderate  number  of  terms  of  such  a  series,  which  would  commence 


106  THE  FUNCTION  <j>  AND 


(343) 
whence 


(344) 


Now  it  has  been  proved  in  Chapter  VII  that 


7  -  ^  _2 

(6  ~~  e)   —  ~  ~r~  €P  ' 
n  dep  p 

We  have  therefore 


approximately.  The  order  of  magnitude  of  rj  —  <£0  is  there- 
fore that  of  log  n.  This  magnitude  is  mainly  constant. 
The  order  of  magnitude  of  rj  +  <pQ  —  \  log  n  is  that  of  unity. 
The  order  of  magnitude  of  </>0  ,  and  therefore  of  —  77,  is  that 
of  n.* 

Equation  (338)  gives  for  the  first  approximation 


(1^  =  _£,  (346) 


(*-*>(.-0  =  ^  =  £*,  W 


/ .  __  ,  Y  —  (6  ~  6o)2  =  ^  ^f  (348) 

a€p 

The  members  of  the  last  equation  have  the  order  of  magnitude 
of  n.     Equation  (338)  gives  also  for  the  first  approximation 


de      fi\  ~  \  ^2  /  v€      eo)> 
*  Compare  (289),  (314). 


THE   CANONICAL  DISTRIBUTION.  107 

whence 


This  is  of  the  order  of  magnitude  of  n.* 

It  should  be  observed  that  the  approximate  distribution  of 
the  ensemble  in  energy  according  to  the  'law  of  errors'  is 
not  dependent  on  the  particular  form  of  the  function  of  the 
energy  which  we  have  assumed  for  the  index  of  probability 
(77).  In  any  case,  we  must  have 


(351) 


where  e^+t  is  necessarily  positive.  This  requires  that  it 
shall  vanish  for  e  =  oo  ,  and  also  for  e  =  —  oo  ,  if  this  is  a  possi- 
ble value.  It  has  been  shown  in  the  last  chapter  that  if  e  has 
a  (finite)  least  possible  value  (which  is  the  usual  case)  and 
n  >  2,  e*  will  vanish  for  that  least  value  of  e.  In  general 
therefore  77  +  <£  will  have  a  maximum,  which  determines  the 
most  probable  value  of  the  energy.  If  we  denote  this  value 
by  e0>  and  distinguish  the  corresponding  values  of  the  func- 
tions of  the  energy  by  the  same  suffix,  we  shall  have 


-a 

The  probability  that  an  unspecified  system  of  the  ensemble 

*  We  shall  find  hereafter  that  the  equation 


is  exact  for  any  value  of  n  greater  than  2,  and  that  the  equation 


fd(f>      IV  __      <^0 
\d*      ®)  '      rf? 
is  exact  for  any  value  of  n  greater  than  4. 


108  THE  FUNCTION  <£  AND 

falls  within  any  given  limits  of  energy  (e'  and  e")  is  repre- 
sented by 


f 


e^de. 


If  we  expand  77  and  <£  in  ascending  powers  of  e  —  e0,  without 
going  beyond  the  squares,  the  probability  that  the  energy  falls 
within  the  given  limits  takes  the  form  of  the  « law  of  errors '  — 


de.  (353) 

i/ 

This  gives 


We  shall  have  a  close  approximation  in  general  when  the 
quantities  equated  in  (355)  are  very  small,  i.  e.,  when 

is  very  great.  Now  when  n  is  very  great,  —  d*$/de*  is  of  the 
same  order  of  magnitude,  and  the  condition  that  (356)  shall 
be  very  great  does  not  restrict  very  much  the  nature  of  the 
function  77. 

We  may  obtain  other  properties  pertaining  to  average  values 
in  a  canonical  ensemble  by  the  method  used  for  the  average  of 
d<j>/de.  Let  u  be  any  function  of  the  energy,  either  alone  or 
with  ®  and  the  external  coordinates.  The  average  value  of  u 
in  the  ensemble  is  determined  by  the  equation 

6=00      4,-e 

/- — -  +  4> 
ue    e          de.  (357) 

F=0 


THE   CANONICAL  DISTRIBUTION.  109 

Now  we  have  identically 


Therefore,  by  the  preceding  equation 


If  we  set  u  =  1,  (a  value  which  need  not  be  excluded,)  the 
second  member  of  this  equation  vanishes,  as  shown  on  page 
101,  if  n  >  2,  and  we  get 

^  =  i,  (360) 

as  before.  It  is  evident  from  the  same  considerations  that  the 
second  member  of  (359)  will  always  vanish  if  n  >  2,  unless  u 
becomes  infinite  at  one  of  the  limits,  in  which  case  a  more  care- 
ful examination  of  the  value  of  the  expression  will  be  necessary. 
To  facilitate  the  discussion  of  such  cases,  it  will  be  convenient 
to  introduce  a  certain  limitation  in  regard  to  the  nature  of  the 
system  considered.  We  have  necessarily  supposed,  in  all  our 
treatment  of  systems  canonically  distributed,  that  the  system 
considered  was  such  as  to  be  capable  of  the  canonical  distri- 
bution with  the  given  value  of  the  modulus.  We  shall  now 
suppose  that  the  system  is  such  as  to  be  capable  of  a  canonical 
distribution  with  any  (finite)  f  modulus.  Let  us  see  what 
cases  we  exclude  by  this  last  limitation. 

*  A  more  general  equation,  which  is  not  limited  to  ensembles  canonically 
distributed,  is 

^  +  M^4.M^-  \uef¥*~\*=*> 
df      U  de      U  de  ~  I"*       J  F=0 

where  t\  denotes,  as  usual,  the  index  of  probability  of  phase. 

t  The  term  finite  applied  to  the  modulus  is  intended  to  exclude  the  value 
zero  as  well  as  infinity. 


110  THE  FUNCTION  0  AND 

The  impossibility  of  a  canonical  distribution  occurs  when 
the  equation 


e  e 


e  =          e 

s*     —  l-j-0 

=J   e   '       de  (361) 

F=0 


fails  to  determine  a  finite  value  for  ^.  Evidently  the  equation 
cannot  make  ty  an  infinite  positive  quantity,  the  impossibility 
therefore  occurs  when  the  equation  makes  ty  =  —  oo  .  Now 
we  get  easily  from  (191) 


If  the  canonical  distribution  is  possible  for  any  values  of  ®, 
we  can  apply  this  equation  so  long  as  the  canonical  distribu- 
tion is  possible.  The  equation  shows  that  as  ®  is  increased 
(without  becoming  infinite)  —  ty  cannot  become  infinite  unless 
6  simultaneously  becomes  infinite,  and  that  as  O  is  decreased 
(without  becoming  zero)  —  ^  cannot  become  infinite  unless 
simultaneously  e  becomes  an  infinite  negative  quantity.  The 
corresponding  cases  in  thermodynamics  would  be  bodies  which 
could  absorb  or  give  out  an  infinite  amount  of  heat  without 
passing  certain  limits  of  temperature,  when  no  external  work 
is  done  in  the  positive  or  negative  sense.  Such  infinite  values 
present  no  analytical  difficulties,  and  do  not  contradict  the 
general  laws  of  mechanics  or  of  thermodynamics,  but  they 
are  quite  foreign  to  our  ordinary  experience  of  nature.  In 
excluding  such  cases  (which  are  certainly  not  entirely  devoid 
of  interest)  we  do  not  exclude  any  which  are  analogous  to 
any  actual  cases  in  thermodynamics. 

We  assume  then  that  for  any  finite  value  of  ®  the  second 
member  of  (361)  has  a  finite  value. 

When  this  condition  is  fulfilled,  the  second  member  of 
(359)  will  vanish  for  u  =  e~+  V.  For,  if  we  set  6'  =  26, 

?  ___!    €  _ f  _  ^     ¥ 

F  =  0  V  =  0 


THE   CANONICAL  DISTRIBUTION.  Ill 

where  tyr  denotes  the  value  of  ^  for  the  modulus  ®'.  Since 
the  last  member  of  this  formula  vanishes  for  e  =  oo ,  the 
less  value  represented  by  the  first  member  must  also  vanish 
for  the  same  value  of  e.  Therefore  the  second  member  of 
(359),  which  differs  only  by  a  constant  factor,  vanishes  at 
the  upper  limit.  The  case  of  the  lower  limit  remains  to  be 
considered.  Now 


The  second  member  of  this  formula  evidently  vanishes  for 
the  value  of  e,  which  gives  V  —  0,  whether  this  be  finite  or 
negative  infinity.  Therefore,  the  second  member  of  (359) 
vanishes  at  the  lower  limit  also,  and  we  have 


V 


or  e      V=®.  (362) 

This  equation,  which  is  subject  to  no  restriction  in  regard  to 
the  value  of  n,  suggests  a  connection  or  analogy  between  the 
function  of  the  energy  of  a  system  which  is  represented  by 
iT^  V  and  the  notion  of  temperature  in  thermodynamics.  We 
shall  return  to  this  subject  in  Chapter  XIV. 

If  n  >  2,  the  second  member  of  (359)  may  easily  be  shown 
to  vanish  for  any  of  the  following  values  of  u  viz.  :  </>,  e^,  e, 
e"*,  where  m  denotes  any  positive  number.  It  will  also 
vanish,  when  n  >  4,  for  u  =  dfyde,  and  when  n  >  2  h  for 
u  =  e-*  dhV/d^.  When  the  second  member  of  (359)  van- 
ishes, and  n  >  2,  we  may  write 


We  thus  obtain  the  following  equations  : 
If  n  >  2, 


(364) 


112 


THE  FUNCTION  </>  AND 


or 


If  w  >  4, 


If  n 


®2 


-<t>dhVd<f>       1    - 
6      -d?-fc-®6 


e   '  -Tjr-j —  e 
de1  ae 


or 


(368) 
t(369) 


(370) 


whence  "  ^-  =  ^. 

Giving  A  the  values  1,  2,  3,  etc.,  we  have 


as  already  obtained.    Also 


*  This  equation  may  also  be  obtained  from  equations  (252)  and  (321). 
Compare  also  equation  (349)  which  was  derived  by  an  approximative  method, 
t  Compare  equation  (360),  obtained  by  an  approximative  method. 


THE   CANONICAL  DISTRIBUTION.  113 

If  Vq  is  a  continuous  increasing  function  of  eg,  commencing 
with  Vq  =  0,  the  average  value  in  a  canonical  ensemble  of  any 
function  of  e^,  either  alone  or  with  the  modulus  and  the  exter- 
nal coordinates,  is  given  by  equation  (275),  which  is  identical 
with  (357)  except  that  e,  $,  and  \jr  have  the  suffix  (  )ff.  The 
equation  may  be  transformed  so  as  to  give  an  equation  iden- 
tical with  (359)  except  for  the  suffixes.  If  we  add  the  same 
suffixes  to  equation  (361),  the  finite  value  of  its  members  will 
determine  the  possibility  of  the  canonical  distribution. 

From  these  data,  it  is  easy  to  derive  equations  similar  to 
(360),  (362)-(372),  except  that  the  conditions  of  their  valid- 
ity must  be  differently  stated.  The  equation 


requires  only  the  condition  already  mentioned  with  respect  to 
Vq.  This  equation  corresponds  to  (362),  which  is  subject  to 
no  restriction  with  respect  to  the  value  of  n.  We  may  ob- 
serve, however,  that  V  will  always  satisfy  a  condition  similar 
to  that  mentioned  with  respect  to  Vr 

If  Vq  satisfies  the  condition  mentioned,  and  e^  a  similar 
condition,  i.  e.,  if  e^i  is  a  continuous  increasing  function  of  e3, 
commencing  with  the  value  (^  =  0,  equations  will  hold  sim- 
ilar to  those  given  for  the  case  when  n  >  2,  viz.,  similar  to 
(360),  (364)-(368).  Especially  important  is 


deq  ~®' 

If  Vq,  6*4  (or  dVq/d€q),  d?Vq/de*  all  satisfy  similar  conditions, 
we  shall  have  an  equation  similar  to  (369),  which  was  subject 
to  the  condition  n  >  4.  And  if  cPVqjdef  also  satisfies  a 
similar  condition,  we  shall  have  an  equation  similar  to  (372), 
for  which  the  condition  was  n  >  6.  Finally,  if  Vq  and  h  suc- 
cessive differential  coefficients  satisfy  conditions  of  the  kind 
mentioned,  we  shall  have  equations  like  (370)  and  (371)  for 
which  the  condition  was  n  >  2  h. 

8 


114  THE  FUNCTION  <£. 

These  conditions  take  the  place  of  those  given  above  relat- 
ing to  n.  In  fact,  we  might  give  conditions  relating  to  the 
differential  coefficients  of  F",  similar  to  those  given  relating  to 
the  differential  coefficients  of  Vq,  instead  of  the  conditions 
relating  to  n,  for  the  validity  of  equations  (360),  (363)-(372). 
This  would  somewhat  extend  the  application  of  the  equations. 


CHAPTER  X. 

ON  A    DISTRIBUTION  IN    PHASE    CALLED  MICROCANONI- 
CAL  IN  WHICH  ALL  THE  SYSTEMS   HAVE 
THE   SAME  ENERGY. 

AN  important  case  of  statistical  equilibrium  is  that  in  which 
all  systems  of  the  ensemble  have  the  same  energy.  We  may 
arrive  at  the  notion  of  a  distribution  which  will  satisfy  the 
necessary  conditions  by  the  following  process.  We  may 
suppose  that  an  ensemble  is  distributed  with  a  uniform  den- 
sity-in-phase  between  two  limiting  values  of  the  energy,  e'  and 
e",  and  with  density  zero  outside  of  those  limits.  Such  an 
ensemble  is  evidently  in  statistical  equilibrium  according  to 
the  criterion  in  Chapter  IV,  since  the  density-in-phase  may  be 
regarded  as  a  function  of  the  energy.  By  diminishing  the 
difference  of  e'  and  e",  we  may  diminish  the  differences  of 
energy  in  the  ensemble.  The  limit  of  this  process  gives  us 
a  permanent  distribution  in  which  the  energy  is  constant. 

We  should  arrive  at  the  same  result,  if  we  should  make  the 
density  any  function  of  the  energy  between  the  limits  e'  and 
e",  and  zero  outside  of  those  limits.  Thus,  the  limiting  distri- 
bution obtained  from  the  part  of  a  canonical  ensemble 
between  two  limits  of  energy,  when  the  difference  of  the 
limiting  energies  is  indefinitely  diminished,  is  independent  of 
the  modulus,  being  determined  entirely  by  the  energy,  and 
is  identical  with  the  limiting  distribution  obtained  from  a 
uniform  density  between  limits  of  energy  approaching  the 
same  value. 

We  shall  call  the  limiting  distribution  at  which  we  arrive 
by  this  process  microcanonical. 

We  shall  find  however,  in  certain  cases,  that  for  certain 
values  of  the  energy,  viz.,  for  those  for  which  e*  is  infinite, 


116        A   PERMANENT  DISTRIBUTION  IN  WHICH 

this  process  fails  to  define  a  limiting  distribution  in  any  such 
distinct  sense  as  for  other  values  of  the  energy.  The  difficulty 
is  not  in  the  process,  but  in  the  nature  of  the  case,  being 
entirely  analogous  to  that  which  we  meet  when  we  try  to  find 
a  canonical  distribution  in  cases  when  ^  becomes  infinite. 
We  have  not  regarded  such  cases  as  affording  true  examples 
of  the  canonical  distribution,  and  we  shall  not  regard  the  cases 
in  which  e^  is  infinite  as  affording  true  examples  of  the  micro- 
canonical  distribution.  We  shall  in  fact  find  as  we  go  on  that 
in  such  cases  our  most  important  formulae  become  illusory. 

The  use  of  formulae  relating  to  a  canonical  ensemble  which 
contain  e^de  instead  of  dpl . . .  dqn,  as  in  the  preceding  chapters, 
amounts  to  the  consideration  of  the  ensemble  as  divided  into 
an  infinity  of  microcanonical  elements; 

From  a  certain  point  of  view,  the  microcanonical  distribution 
may  seem  more  simple  than  the  canonical,  and  it  has  perhaps 
been  more  studied,  and  been  regarded  as  more  closely  related 
to  the  fundamental  notions  of  thermodynamics.  To  this  last 
point  we  shall  return  in  a  subsequent  chapter.  It  is  sufficient 
here  to  remark  that  analytically  the  canonical  distribution  is 
much  more  manageable  than  the  microcanonical. 

We  may  sometimes  avoid  difficulties  which  the  microcanon- 
ical distribution  presents  by  regarding  it  as  the  result  of  the 
following  process,  which  involves  conceptions  less  simple  but 
more  amenable  to  analytical  treatment.  We  may  suppose  an 
ensemble  distributed  with  a  density  proportional  to 


where  &>  and  e1  are  constants,  and  then  diminish  indefinitely 
the  value  of  the  constant  &>.  Here  the  density  is  nowhere 
zero  until  we  come  to  the  limit,  but  at  the  limit  it  is  zero  for 
all  energies  except  e'.  We  thus  avoid  the  analytical  compli- 
cation of  discontinuities  in  the  value  of  the  density,  which 
require  the  use  of  integrals  with  inconvenient  limits. 

In  a  microcanonical  ensemble  of  systems  the  energy  (e)  is 
constant,  but  the  kinetic  energy  (e^)  and  the  potential  energy 


ALL  SYSTEMS  HAVE   THE  SAME  ENERGY.        117 

(eq)  vary  in  the  different  systems,  subject  of  course  to  the  con- 
dition 

€p  -f  eq  =  e  =  constant.  (373) 

Our  first  inquiries  will  relate  to  the  division  of  energy  into 
these  two  parts,  and  to  the  average  values  of  functions  of  ep 
and  eq. 

We  shall  use  the  notation  y\  6  to  denote  an  average  value  in 
a  microcanonical  ensemble  of  energy  e.  An  average  value 
in  a  canonical  ensemble  of  modulus  (D,  which  has  hitherto 
been  denoted  by  M,  we  shall  in  this  chapter  denote  by  '^@,  to 
distinguish  more  clearly  the  two  kinds  of  averages. 

The  extension-in-phase  within  any  limits  which  can  be  given 
in  terms  of  ep  and  eq  may  be  expressed  in  the  notations  of  the 
preceding  chapter  by  the  double  integral 

*dVpdVq 

taken  within  those  limits.  If  an  ensemble  of  systems  is  dis- 
tributed within  those  limits  with  a  uniform  density-in-phase, 
the  average  value  in  the  ensemble  of  any  function  (u)  of  the 
kinetic  and  potential  energies  will  be  expressed  by  the  quotient 

of  integrals 

/»  r 

udVpdVq 


dVpdVq 


Since  d  Vp  =  e^p  dep,  and  dep  =  de  when  eq  is  constant,  the 
expression  may  be  written 


To  get  the  average  value  of  u  in  an  ensemble  distributed 
microcanonically  with  the  energy  6,  we  must  make  the  in- 
tegrations cover  the  extension-in-phase  between  the  energies 
e  and  e  +  de.  This  gives 


118        A   PERMANENT  DISTRIBUTION  IN  WHICH 


de\ueVpdVq 


vq=o 

But  by  (299)  the  value  of  the  integral  in  the  denominator 
is  e^.     We  have  therefore 


(374) 


where  e^p  and  Vq  are  connected  by  equation  (373),  and  w,  if 
given  as  function  of  ep,  or  of  ep  and  eq,  becomes  in  virtue  of 
the  same  equation  a  function  of  eq  alone. 

We  shall  assume  that  e^  has  a  finite  value.  If  n  >  1,  it  is 
evident  from  equation  (305)  that  e^  is  an  increasing  function 
of  e,  and  therefore  cannot  be  infinite  for  one  value  of  e  without 
being  infinite  for  all  greater  values  of  e,  which  would  make 
—  ty  infinite.*  When  n  >  1,  therefore,  if  we  assume  that  e^ 
is  finite,  we  only  exclude  such  cases  as  we  found  necessary 
to  exclude  in  the  study  of  the  canonical  distribution.  But 
when  n  =  1,  cases  may  occur  in  which  the  canonical  distribu- 
tion is  perfectly  applicable,  but  in  which  the  formulae  for  the 
microcanonical  distribution  become  illusory,  for  particular  val- 
ues of  e,  on  account  of  the  infinite  value  of  e^.  Such  failing 
cases  of  the  microcanonical  distribution  for  particular  values 
of  the  energy  will  not  prevent  us  from  regarding  the  canon- 
ical ensemble  as  consisting  of  an  infinity  of  microcanonical 
ensembles,  f 

*  See  equation  (322). 

t  An  example  of  the  failing  case  of  the  microcanonical  distribution  is 
afforded  by  a  material  point,  under  the  influence  of  gravity,  and  constrained 
to  remain  in  a  vertical  circle.  The  failing  case  occurs  when  the  energy  is 
just  sufficient  to  carry  the  material  point  to  the  highest  point  of  the  circle. 

It  will  be  observed  that  the  difficulty  is  inherent  in  the  nature  of  the  case, 
and  is  quite  independent  of  the  mathematical  formulae.  The  nature  of  the 
difficulty  is  at  once  apparent  if  we  try  to  distribute  a  finite  number  of 


ALL  SYSTEMS  HAVE   THE  SAME  ENERGY.        119 
From  the  last  equation,  with  (298),  we  get 

=  e~*  V.  (375) 

But  by  equations  (288)  and  (289) 

•-V,-?*.  (376) 

Therefore  

e~*  V—  e~  P  "Pjj  e  =  -  ep\e .  (377) 

Again,  with  the  aid  of  equation  (301),  we  get 

=  £»  (378) 


Vq=0 

if  n  >  2.     Therefore,  by  (289) 


These  results  are  interesting  on  account  of  the  relations  of 
the  functions  e~$  V  and  -^  to  the  notion  of  temperature  in 

thermodynamics,  —  a  subject  to  which  we  shall  return  here- 
after. They  are  particular  cases  of  a  general  relation  easily 
deduced  from  equations  (306),  (374),  (288)  and  (289).  We 
have 


•         '  '        r    : ,       .    w  < 


f* 

=J 


The  equation  may  be  written 

€g=< 


material  points  with  this  particular  value  of  the  energy  as  nearly  as  possible 
in  statistical  equilibrium,  or  if  we  ask :  What  is  the  probability  that  a  point 
taken  at  random  from  an  ensemble  in  statistical  equilibrium  with  this  value 
of  the  energy  will  be  found  in  any  specified  part  of  the  circle? 


120        A   PERMANENT  DISTRIBUTION  IN   WHICH 
We  have  therefore 


if  h  <  J-  n  +  1.     For  example,  when  w  is  even,  we  may  make 
A  =  i-  n,  which  gives,  with  (307), 


1-2 


(381) 


Since  any  canonical  ensemble  of  systems  may  be  regarded 
as  composed  of  microcanonical  ensembles,  if  any  quantities 
u  and  v  have  the  same  average  values  in  every  microcanonical 
ensemble,  they  will  have  the  same  values  in  every  canonical 
ensemble.  To  bring  equation  (380)  formally  under  this  rule, 
we  may  observe  that  the  first  member  being  a  function  of  e  is 
a  constant  value  in  a  microcanonical  ensemble,  and  therefore 
identical  with  its  average  value.  We  get  thus  the  general 
equation 


.-*£? 


if  h  <  J  n  +  1.*     The  equations 

.          9  _ 

(383) 


may  be  regarded  as  particular  cases  of  the  general  equation. 
The  last  equation  is  subject  to  the  condition  that  n  >  2. 

The  last  two  equations  give  for  a  canonical  ensemble, 
x       if  n  >  2, 

(l-|)^leV^]0-l.  (385) 

The  corresponding  equations  for  a  microcanonical  ensemble 
give,  if  n  >  2, 

\l    1  A    1      '     _1|  ^V*  /OQ£\ 

I1  -  =  I  W«  V>  =  ^wTF'  (386) 


See  equation  (292). 


ALL  SYSTEMS  HAVE   THE  SAME  ENERGY.        121 

which   shows  that   d$   dlog  V  approaches  the  value  unity 
when  n  is  very  great. 

If  a  system  consists  of  two  parts,  having  separate  energies, 
we  may  obtain  equations  similar  in  form  to  the  preceding, 
which  relate  to  the  system  as  thus  divided.*  We  shall 
distinguish  quantities  rekting  to  the  parts  by  letters  with 
suffixes,  the  same  letters  without  suffixes  relating  to  the 
whole  system.  The  extension-in-phase  of  the  whole  system 
within  any  given  limits  of  the  energies  may  be  represented  by 
the  double  integral 


taken  within  those  limits,  as  appears  at  once  from  the  defini- 
tions of  Chapter  VIII.  In  an  ensemble  distributed  with 
uniform  density  within  those  limits,  and  zero  density  outside, 
the  average  value  of  any  function  of  e1  and  ea  is  given  by  the 
quotient 


which  may  also  be  written  f 


If  we  make  the  limits  of  integration  e  and  e  +  de,  we  get  the 

*  If  this  condition  is  rigorously  fulfilled,  the  parts  will  have  no  influence 
on  each  other,  and  the  ensemble  formed  by  distributing  the  whole  micro- 
canonically  is  too  arbitrary  a  conception  to  have  a  real  interest.  The  prin- 
cipal interest  of  the  equations  which  we  shall  obtain  will  be  in  cases  in 
which  the  condition  is  approximately  fulfilled.  But  for  the  purposes  of  a 
theoretical  discussion,  it  is  of  course  convenient  to  make  such  a  condition 
absolute.  Compare  Chapter  IV,  pp.  35  ff.,  where  a  similar  condition  is  con- 
sidered in  connection  with  canonical  ensembles. 

t  Where  the  analytical  transformations  are  identical  in  form  with  those 
on  the  preceding  pages,  it  does  not  appear  necessary  to  give  all  the  steps 
with  the  same  detail. 


122        A   PERMANENT  DISTRIBUTION  IN   WHICH 

average  value  of  u  in  an  ensemble  in  which  the  whole  system 
is  microcanonically  distributed  in  phase,  viz., 


(387) 


where  fa  and  V2  are  connected  by  the  equation 

€i  +  €2  =  constant  =  e,  (388) 

and  u,  if  given  as  function  of  ei  ,  or  of  ei  and  e2  ,  becomes  in 

virtue  of  the  same  equation  a  function  of  e2  alone.* 

Thus 


Je  =  e+  J  F!  rf  F2  ,  (389) 

(390) 


This  requires  a  similar  relation  for  canonical  averages 

©  =  e~+  V\e  =  e^rje  =  e~+*V\*  .  (391) 

Again 

e2=e 

SB  =e-*f  ^V'rfF,.  (392) 

del  |e  J    del 

F^O 

But  if  w:  >  2,  «*i  vanishes  for  Fj  =  0,f  and 


.  (393) 

de 


Hence,  if  n^  >  2,  and  w2  >  2, 

d<f>  _  dfal    _  dfa\  /qq  .. 

^e  ~  ^  |f  ~  dez  \f 

*  In  the  applications  of  the  equation  (387),  we  cannot  obtain  all  the  results 
corresponding  to  those  which  we  have  obtained  from  equation  (374),  because 
<t>p  is  a  known  function  of  ep,  while  fa  must  be  treated  as  an  arbitrary  func- 
tion of  €j  ,  or  nearly  so. 

t  See  Chapter  VIII,  equations  (306)  and  (316). 


ALL  SYSTEMS  HAVE   THE  SAME  ENERGY.        123 


and  s « 5l   =^\   =  ^  •  (395) 

©       de  |0       rfej  J0       rfe2  |e 

We  have  compared  certain  functions  of  the  energy  of  the 
whole  system  with  average  values  of  similar  functions  of 
the  kinetic  energy  of  the  whole  system,  and  with  average 
values  of  similar  functions  of  the  whole  energy  of  a  part  of 
the  system.  We  may  also  compare  the  same  functions  with 
average  values  of  the  kinetic  energy  of  a  part  of  the  system. 

We  shall  express  the  total,  kinetic,  and  potential  energies  of 
the  whole  system  by  e,  ep,  and  eg,  and  the  kinetic  energies  of  the 
parts  by  e^,  and  e2p.  These  kinetic  energies  are  necessarily  sep- 
arate :  we  need  not  make  any  supposition  concerning  potential 
energies.  The  extension-in-phase  within  any  limits  which  can 
be  expressed  in  terms  of  eg,  e^,  ezp  may  be  represented  in  the 
notations  of  Chapter  VIII  by  the  triple  integral 


taken  within  those  limits.  And  if  an  ensemble  of  systems  is 
distributed  with  a  uniform  density  within  those  limits,  the 
average  value  of  any  function  of  eq,  e^,  e^  will  be  expressed 
by  the  quotient 


fffue^ded  VZpd  Vq 


or 


To  get  the  average  value  of  u  for  a  microcanonical  distribu- 
tion, we  must  make  the  limits  e  and  e  +  de.  The  denominator 
in  this  case  becomes  e^  de,  and  we  have 

C2p=C— Cq 

(396) 


124        A   PERMANENT  DISTRIBUTION  IN  WHICH 
where  0^,  V2P,  and  Vq  are  connected  by  the  equation 

€ip  +  €2p  +  eq  =  constant  =  e. 
Accordingly 


J  VlpdV2p  dVq  =  e-*  V,      (397) 


and  we  may  write 


;r  2   ,  2    j  /onON 

2p|6  =  — e^l€  =  -€^|e,      (398) 


and 


O  f) 

r    \     _  _  ^ — I     __        ^ — |       ('399') 


Again,  if  wx  >  2, 


C9=€        (ft  (ft 

~*    C^'jir         "*"** 

=  e  J  ^dFi=«  ir* 


Hence,  if  ^  >  2,  and  w2  >  2, 


_  2p          _    f  i  1  N          -1)     _   /I  w  -IN  f     -11 

de  ~de~l       '*      "~  ''    p    ^€  ~  ^      ~~    '    p    '€ 


We  cannot  apply  the  methods  employed  in  the  preceding 
pages  to  the  microcanonical  averages  of  the  (generalized) 
forces  Av  Ay,  etc.,  exerted  by  a  system  on  external  bodies, 
since  these  quantities  are  not  functions  of  the  energies,  either 
kinetic  or  potential,  of  the  whole  or  any  part  of  the  system. 
We  may  however  use  the  method  described  on  page  116. 


ALL   SYSTEMS  HAVE   THE  SAME  ENERGY.        125 

Let  us  imagine  an  ensemble  of  systems  distributed  in  phase 
according  to  the  index  of  probability 

(e  -  c'V 


where  ef  is  any  constant  which  is  a  possible  value  of  the 
energy,  except  only  the  least  value  which  is  consistent  with 
the  values  of  the  external  coordinates,  and  c  and  o>  are  other 
constants.  We  have  therefore 


all 

c— • 


e,        w      dpl  .  .  .  dqn  —  1,  (403) 

phases 


or  e     =...e  dPl  .  .  .  dqn,  (404) 

phases 


_c  |  g 

or  again  e     =  C    e      ^          de.  (405) 


From  (404)  we  have 

all 


phases 

=  00 


,  j 

^  (406) 


where  H7ie  denotes  the  average  value  of  A1  in  those  systems 
of  the  ensemble  which  have  any  same  energy  e.  (This 
is  the  same  thing  as  the  average  value  of  A  l  in  a  microcanoni- 
cal  ensemble  of  energy  e.)  The  validity  of  the  transformation 
is  evident,  if  we  consider  separately  the  part  of  each  integral 
which  lies  between  two  infimtesimally  differing  limits  of 
energy.  Integrating  by  parts,  we  get 


126        A  PERMANENT  DISTRIBUTION  IN  WHICH 


Jr=o 

(*-O, 

•j  .  v  — '  •  -•  "j~Q> 

F=0  ^  / 

Differentiating  (405),  we  get 

€=00  (f-O2  (*~O2 

de-*        rdcj>       —rf—+*   _        /  -  ~~rf~+<t> dea\ 

T—  =    I  -£-  e  de—[e  — ±  } 

^  da^       J  dc^  \  ddij 

where  ea  denotes  the  least  value  of  e  consistent  with  the  exter- 
nal coordinates.  The  last  term  in  this  equation  represents  the 
part  of  de~c  jda^  which  is  due  to  the  variation  of  the  lower 
limit  of  the  integral.  It  is  evident  that  the  expression  in  the 
brackets  will  vanish  at  the  upper  limit.  At  the  lower  limit, 
at  which  ep  =  0,  and  eq  has  the  least  value  consistent  with  the 
external  coordinates,  the  average  sign  on  ^]6  is  superfluous, 
as  there  is  but  one  value  of  A1  which  is  represented  by 
—  dea/dar  Exceptions  may  indeed  occur  for  particular  values 
of  the  external  coordinates,  at  which  dejda^  receive  a  finite 
increment,  and  the  formula  becomes  illusory.  Such  particular 
values  we  may  for  the  moment  leave  out  of  account.  The 
last  term  of  (408)  is  therefore  equal  to  the  first  term  of  the 
second  member  of  (407).  (We  may  observe  that  both  vanish 
when  n  >  2  on  account  of  the  factor  e$.) 
We  have  therefore  from  these  equations 


F=0 


or 


That  is :  the  average  value  in  the  ensemble  of  the  quantity 
represented  by  the  principal  parenthesis  is  zero.     This  must 


ALL  SYSTEMS  HAVE   THE  SAME  ENERGY.        127 

be  true  for  any  value  of  «.  If  we  diminish  o>,  the  average 
value  of  the  parenthesis  at  the  limit  when  «  vanishes  becomes 
identical  with  the  value  for  e  =  e'.  But  this  may  be  any  value 
of  the  energy,  except  the  least  possible.  We  have  therefore 


unless  it  be  for  the  least  value  of  the  energy  consistent  with 
the  external  coordinates,  or  for  particular  values  of  the  ex- 
ternal coordinates.  But  the  value  of  any  term  of  this  equa- 
tion as  determined  for  particular  values  of  the  energy  and 
of  the  external  coordinates  is  not  distinguishable  from  its 
value  as  determined  for  values  of  the  energy  and  external 
coordinates  indefinitely  near  those  particular  values.  The 
equation  therefore  holds  without  limitation.  Multiplying 
by  e*,  we  get 


=  e== 


The  integral  of  this  equation  is 


where  Fl  is  a  function  of  the  external  coordinates.  We  have 
an  equation  of  this  form  for  each  of  the  external  coordinates. 
This  gives,  with  (266),  for  the  complete  value  of  the  differen- 
tial of  V 

dV=e*de  +  (/Ale  -  ty  da,,  +  (e+^k-F^dat  +  etc.,  (413) 
or 

d  V=  £  (de  +  !ZT|e  dai  +  3^]e  daz  +  etc.)  —  Fldal  —  Fz  daz  —  etc. 

(414) 

To  determine  the  values  of  the  functions  Fl  ,  Fz  ,  etc.,  let 
us  suppose  a-L  ,  «2  ,  etc.  to  vary  arbitrarily,  while  e  varies  so 
as  always  to  have  the  least  value  consistent  with  the  values 
of  the  external  coordinates.  This  will  make  V=  0,  and 
dV=  0.  If  7i  <  2,  we  shall  have  also  e*  =  0,  which  will 
give 

JF1  =  0,    -F2  =  0,    etc.  (415) 


128  THE  MICROCANONICAL  DISTRIBUTION. 

The  result  is  the  same  for  any  value  of  n.  For  in  the  varia- 
tions considered  the  kinetic  energy  will  be  constantly  zero, 
and  the  potential  energy  will  have  the  least  value  consistent 
with  the  external  coordinates.  The  condition  of  the  least 
possible  potential  energy  may  limit  the  ensemble  at  each  in- 
stant to  a  single  configuration,  or  it  may  not  do  so  ;  but  in  any 
case  the  values  of  A1  ,  Av  etc.  will  be  the  same  at  each  instant 
for  all  the  systems  of  the  ensemble,*  and  the  equation 

de  +  Al  da^  -f  Az  daz  +  etc.  =  0 

will  hold  for  the  variations  considered.  Hence  the  functions 
F^  ,  F%  ,  etc.  vanish  in  any  case,  and  we  have  the  equation 

d  V=  e*de  +  e*  Z^d^  +  e+~Z^dat  +  etc.,  (416) 


de  +  ~A\,dal  +  Z^Lrfa2  +  etc. 
or  dlogV=;-  _0      '6  -  (417) 

or  again 

de  =  e~*  V  d  log  V  -  "27]€  dot  -  lj|e  da2  -  etc.          (418) 

It  will  be  observed  that  the  two  last  equations  have  the  form 
of  the  fundamental  differential  equations  of  thermodynamics, 
er-^V  corresponding  to  temperature  and  log  V  to  entropy. 
We  have  already  observed  properties  of  &"*>  V  suggestive  of  an 
analogy  with  temperature,  f  The  significance  of  these  facts 
will  be  discussed  in  another  chapter. 

The  two  last  equations  might  be  written  more  simply 

de  +  37|€  dct!  +  Af€  daz  +  etc. 

™  *    —  '  -  —  7  -  j 

er-4 
de  =  e~^  d  V  —  "37)€  da^  —  ~A^\€  da2  —  etc., 

and  still  have  the  form  analogous  to  the  thermodynamic 
equations,  but  e~^  has  nothing  like  the  analogies  with  tempera- 
ture which  we  have  observed  in  e~^  V. 

*  This  statement,  as  mentioned  before,  may  have  exceptions  for  particular 
values  of  the  external  coordinates.  This  will  not  invalidate  the  reasoning, 
which  has  to  do  with  varying  values  of  the  external  coordinates. 

t  See  Chapter  IX,  page  111  ;  also  this  chapter,  page  119. 


CHAPTER  XI. 

MAXIMUM  AND  MINIMUM  PROPERTIES  OF  VARIOUS  DIS- 
TRIBUTIONS IN  PHASE. 

IN  the  following  theorems  we  suppose,  as  always,  that  the 
systems  forming  an  ensemble  are  identical  in  nature  and  in 
the  values  of  the  external  coordinates,  which  are  here  regarded 
as  constants. 

Theorem  I.  If  an  ensemble  of  systems  is  so  distributed  in 
phase  that  the  index  of  probability  is  a  function  of  the  energy, 
the  average  value  of  the  index  is  less  than  for  any  other  distri- 
bution in  which  the  distribution  in  energy  is  unaltered. 

Let  us  write  TJ  for  the  index  which  is  a  function  of  the 
energy,  and  77  +  A??  for  any  other  which  gives  the  same  dis- 
tribution in  energy.  It  is  to  be  proved  that 

all  all 

J*.  .  .  J*  (i,  +  Ar,)  e"**1  dPl...  dqn  >f.  .  .  Jr?  6*  dp,...  dqn  ,  (419) 

pliases  phases 

where  ??  is  a  function  of  the  energy,  and  A?;  a  function  of  the 
phase,  which  are  subject  to  the  conditions  that 

all  all 

J.  .  .  Je^4"  dp,...  dqn  =  f.  .  .  J>  d&...  dyn  =  1,  (420) 

phases  phases 

and  that  for  any  value  of  the  energy  (e') 


dp,...  dqn  =.  .  .fdpi  ...dqn.      (421) 


Equation  (420)  expresses  the  general  relations  which  -77  and 
77  +  AT;  must  satisfy  in  order  to  be  indices  of  any  distributions, 
and  (421)  expresses  the  condition  that  they  give  the  same 
distribution  in  energy. 


130          MAXIMUM  AND  MINIMUM  PROPERTIES. 

Since  77  is  a  function  of  the  energy,  and  may  therefore  be  re- 
garded as  a  constant  within  the  limits  of  integration  of  (421), 
we  may  multiply  by  T;  under  the  integral  sign  hi  both  mem- 
bers, which  gives 


C 

J. 


71  dp^  .  .  .  dqn. 

U  U  \J 

€=«'  €— e' 

Since  this  is  true  within  the  limits  indicated,  and  for  every 
value  of  e',  it  will  be  true  if  the  integrals  are  taken  for  all 
phases.  We  may  therefore  cancel  the  corresponding  parts  of 
(419),  which  gives 

all 

f A  r,  e1**11  dPl...  dqn  >  0.  (422) 

J 

phases 

But  by  (420)  this  is  equivalent  to 

all 

/.  . .  /  (Ar;eAl7  +  1  —  e^e'dpi . . .  dqn  >  0.         (423) 
tj 

phases 

Now  AT;  e^  +  1  —  e^  is  a  decreasing  function  of  AT;  for  nega- 
tive values  of  AT;,  and  an  increasing  function  of  AT;  for  positive 
values  of  AT;.  It  vanishes  for  AT;  =  0.  The  expression  is 
therefore  incapable  of  a  negative  value,  and  can  have  the  value 
0  only  for  AT;  =  0.  The  inequality  (423)  will  hold  therefore 
unless  AT;  =  0  for  all  phases.  The  theorem  is  therefore 
proved. 

Theorem  II.  If  an  ensemble  of  systems  is  canonically  dis- 
tributed in  phase,  the  average  index  of  probability  is  less  than 
in  any  other  distribution  of  the  ensemble  having  the  same 
average  energy. 

For  the  canonical  distribution  let  the  index  be  (^  —  e)  /  ®, 
and  for  another  having  the  same  average  energy  let  the  index 
be  (t/r  —  e)/0  +  AT;,  where  AT;  is  an  arbitrary  function  of  the 
phase  subject  only  to  the  limitation  involved  in  the  notion  of 
the  index,  that 


MAXIMUM  AND  MINIMUM  PROPERTIES.          131 

all  itr—  f  a11  J'—  € 

/(*      +  AIJ  r      r  — 

.  .  .J  e*  dPl  .  .  .  dqn=J  .  .  .J  e  &   dPl  .  .  .  dqn  =  1, 

phases  phases 

(424) 
and  to  that  relating  to  the  constant  average  energy,  that 

all  —  f  all 


J.  .  .  Je  e"^"4  *  4,,  .  .  .  <*?„  =J  .  .  .  Je  e~e~  fe  .  .  .  <*?..     (425) 

phases  phases 

It  is  to  be  proved  that 


phases 

all 


phases 

Now  in  virtue  of  the  first  condition  (424)  we  may  cancel  the 
constant  term  ^r  /®  in  the  parentheses  in  (426),  and  in  virtue 
of  the  second  condition  (425)  we  may  cancel  the  term  e/O. 
The  proposition  to  be  proved  is  thus  reduced  to 

all  ty~€ 

I  A>7  e  &          dpi  . .  .  dqn  >  0, 

phases 

which  may  be  written,  in  virtue  of  the  condition  (424), 

all  if/— e 

f.  .  .  f  (Ar;  eAl?  +  1  -  /")  e®~  dpi...  dqn  >  0.        (427) 
J       J 

phases 

In  this  form  its  truth  is  evident  for  the  same  reasons  which 
applied  to  (423). 

Theorem  III.  If  ®  is  any  positive  constant,  the  average 
value  in  an  ensemble  of  the  expression  77  -|-  e  /  0  (77  denoting 
as  usual  the  index  of  probability  and  e  the  energy)  is  less  when 
the  ensemble  is  distributed  canonically  with  modulus  ©,  than 
for  any  other  distribution  whatever. 

In  accordance  with  our  usual  notation  let  us  write 
(i/r  —  e)  /  ®  for  the  index  of  the  canonical  distribution.  In  any 
other  distribution  let  the  index  be  (>/r  —  e)/®  +  AT;. 


132          MAXIMUM  AND  MINIMUM  PROPERTIES. 

In  the  canonical  ensemble  rj  +  e  /  ©  has  the  constant  value 
-»|r  /  <s) ;  in  the  other  ensemble  it  has  the  value  A/T  /  ©  -f-  A?/. 
The  proposition  to  be  proved  may  therefore  be  written 


all 


phases 

where 


r/-    ^ 
dPl...dqn=J...Je  e  dPl...d<i,  =  l.     (429) 

phases  phases 

In  virtue  of  this  condition,  since  i/r  /  ®  is  constant,  the  propo- 
sition to  be  proved  reduces  to 

all  j-t 

//»        —^r  +  Af 
...J  A^6    <  cZql...dpn,  (430) 

phases 

where  the  demonstration  may  be  concluded  as  in  the  last 
theorem. 

If  we  should  substitute  for  the  energy  in  the  preceding 
theorems  any  other  function  of  the  phase,  the  theorems,  mu- 
tatis mutandis,  would  still  hold.  On  account  of  the  unique 
importance  of  the  energy  as  a  function  of  the  phase,  the  theo- 
rems as  given  are  especially  worthy  of  notice.  When  the  case 
is  such  that  other  functions  of  the  phase  have  important 
properties  relating  to  statistical  equilibrium,  as  described 
in  Chapter  IV,*  the  three  following  theorems,  which  are 
generalizations  of  the  preceding,  may  be  useful.  It  will  be 
sufficient  to  give  them  without  demonstration,  as  the  principles 
involved  are  in  no  respect  different. 

Theorem  IV.  If  an  ensemble  of  systems  is  so  distributed  in 
phase  that  the  index  of  probability  is  any  function  of  Fv  JP2, 
etc.,  (these  letters  denoting  functions  of  the  phase,)  the  average 
value  of  the  index  is  less  than  for  any  other  distribution  in 
phase  in  which  the  distribution  with  respect  to  the  functions 
Fv  Fv  etc.  is  unchanged. 

*  See  pages  37-41. 


MAXIMUM  AND  MINIMUM  PROPERTIES.          133 

Theorem  V.  If  an  ensemble  of  systems  is  so  distributed 
in  phase  that  the  index  of  probability  is  a  linear  function  of 
Fv  Fv  etc.,  (these  letters  denoting  functions  of  the  phase,)  the 
average  value  of  the  index  is  less  than  for  any  other  distribu- 
tion in  which  the  functions  Fv  F^  etc.  have  the  same  average 
values. 

Theorem  VI.  The  average  value  in  an  ensemble  of  systems 
of  77  +  F  (where  77  denotes  as  usual  the  index  of  probability  and 
F  any  function  of  the  phase)  is  less  when  the  ensemble  is  so 
distributed  that  77  +  F  is  constant  than  for  any  other  distribu- 
tion whatever. 

Theorem  VII.  If  a  system  which  in  its  different  phases 
constitutes  an  ensemble  consists  of  two  parts,  and  we  consider 
the  average  index  of  probability  for  the  whole  system,  and 
also  the  average  indices  for  each  of  the  parts  taken  separately, 
the  sum  of  the  average  indices  for  the  parts  will  be  either  less 
than  the  average  index  for  the  whole  system,  or  equal  to  it, 
but  cannot  be  greater.  The  limiting  case  of  equality  occurs 
when  the  distribution  in  phase  of  each  part  is  independent  of 
that  of  the  other,  and  only  in  this  case. 

Let  the  coordinates  and  momenta  of  the  whole  system  be 

2l  •  •  •  ZifiPl  >  •  •  -Pni   Of  Wnicl1  ft  •  '  •  <lm  Pi  ,  •  •  -Pm  relate  to  °ne 

part  of  the  system,  and  qm+l  ,...<?„,  pm+l , .  .  .  pn  to  the  other. 
If  the  index  of  probability  for  the  whole  system  is  denoted  by 
77,  the  probability  that  the  phase  of  an  unspecified  system  lies 
within  any  given  limits  is  expressed  by  the  integral 

f.  .  .fe*dPl  ...dq,  (431) 

taken  for  those  limits.     If  we  set 

J  .  .  .fa  dpm+l  .  .  .  dpndq^  ...dqn=.  e\  (432) 

where  the  integrations  cover  all  phases  of  the  second  system, 
and 

J.  .  .  JV  dPl  . .  .  dpm  dqi...  dqm  =  e^  (433) 


134          MAXIMUM  AND  MINIMUM  PROPERTIES. 

where  the  integrations  cover  all  phases  of  the  first  system, 
the  integral  (431)  will  reduce  to  the  form 


f  .  .  . 


dp!...  dpmd^  .  .  .  dqm)  (434) 

when  the  limits  can  be  expressed  in  terms  of  the  coordinates 
and  momenta  of  the  first  part  of  the  system.  The  same  integral 
will  reduce  to 


J  .  .  .  J  (?*  dpm+l  ...dpn  dqm+1  ...dqr 


(435) 


when  the  limits  can  be  expressed  in  terms  of  the  coordinates 
and  momenta  of  the  second  part  of  the  system.  It  is  evident 
that  rj1  and  r)2  are  the  indices  of  probability  for  the  two  parts 
of  the  system  taken  separately. 

The  main  proposition  to  be  proved  may  be  written 


f  • 


(436) 


where  the  first  integral  is  to  be  taken  over  all  phases  of  the  first 
part  of  the  system,  the  second  integral  over  all  phases  of  the 
second  part  of  the  system,  and  the  last  integral  over  all  phases 
of  the  whole  system.  Now  we  have 

..%.  =  !,  (437) 

..dqm  =  lt  (438) 

and  * 

where  the  limits  cover  in  each  case  all  the  phases  to  which  the 
variables  relate.  The  two  last  equations,  which  are  in  them- 
selves evident,  may  be  derived  by  partial  integration  from  the 
first. 


J*.  .  .Je^dpm+l  ...dqn  =  l,  (439) 


MAXIMUM  AND  MINIMUM  PROPERTIES.          135 

It  appears  from  the  definitions  of  ^  and  7?2  that  (436)  may 
also  be  written 

f  .  .  .  Cru  endpl  ...dqn  +  J".  .  .  J  ^  e^dpl  ...dqn< 

f...  fa  <&..<<%.,     (440) 

or  f  .  .  .  f  0?  -  >?i  -  in)***!  .  .  .  dqn  >  0, 

where  the  integrations  cover  all  phases.     Adding  the  equation 

...  <?<?*  =  1,  (441) 


f  . 


a 

f.  .  .  C 


which  we  get  by  multiplying  (438)  and  (439),  and  subtract- 
ing (437),  we  have  for  the  proposition  to  be  proved 

all 

J.  .  .J[(,  -  %  -  Tfc)  J  +  «»**  -  e"]  <$*  .  .  .  dqn  >  0.     (442) 

phases 

Let 

U  =  r1  —  r}1  —  r]2.  (443) 

The  main  proposition  to  be  proved  may  be  written 

all 

n  >  0.          (444) 

phases 

This  is  evidently  true  since  the  quantity  in  the  parenthesis  is 
incapable  of  a  negative  value.*  Moreover  the  sign  =  can 
hold  only  when  the  quantity  in  the  parenthesis  vanishes  for 
all  phases,  i.  e.,  when  u  =  0  for  all  phases.  This  makes 
i)  =  tjl  +  ?72  for  all  phases,  which  is  the  analytical  condition 
which  expresses  that  the  distributions  in  phase  of  the  two 
parts  of  the  system  are  independent. 

Theorem  VIII.  If  two  or  more  ensembles  of  systems  which 
are  identical  in  nature,  but  may  be  distributed  differently  in 
phase,  are  united  to  form  a  single  ensemble,  so  that  the  prob- 
ability-coefficient of  the  resulting  ensemble  is  a  linear  function 

*  See  Theorem  I,  where  this  is  proved  of  a  similar  expression. 


136          MAXIMUM  AND  MINIMUM  PROPERTIES. 

of  the  probability-coefficients  of  the  original  ensembles,  the 
average  index  of  probability  of  the  resulting  ensemble  cannot 
be  greater  than  the  same  linear  function  of  the  average  indices 
of  the  original  ensembles.  It  can  be  equal  to  it  only  when 
the  original  ensembles  are  similarly  distributed  in  phase. 

Let  PijP%,  etc.  be  the  probability-coefficients  of  the  original 
ensembles,  and  P  that  of  the  ensemble  formed  by  combining 
them  ;  and  let  N^  ,  -ZV^  ,  etc.  be  the  numbers  of  systems  in  the 
original  ensembles.  It  is  evident  that  we  shall  have 

P  =  elPl  +  c2P2  +  etc.  =  2  (cjPi),  (445) 

where  Ci  =  =-^V>    c2  =  —^,    etc.  (446) 


The  main  proposition  to  be  proved  is  that 


all  all 

/•  •  ./P  log  PdPl . . .  <*?„  ^  s  pi/  •  -/P,  log  P,  ^ . . .  dfcTI 

phases  L       phases  -• 

(447) 

all 

f . .  .  f  [2  (clPl  log  PO  -  P  log  P]  dPl...  dqn  >  0.    (448) 
J       J 


or 

J 

phases 

If  we  set 

ft  =  P!  log  P!  -  P!  log  P  -  P!  +  P 

Q1  will  be  positive,  except  when  it  vanishes  for  P1  =  P.  To 
prove  this,  we  may  regard  Pl  and  P  as  any  positive  quantities. 
Then 


\dPi*JP      PI  ' 

Since  Q1  and  dQ1/dP1  vanish  for  Pl  —  P,  and  the  second 
differential  coefficient  is  always  positive,  Q1  must  be  positive 
except  when  P1  =  P.  Therefore,  if  #2,  etc.  have  similar 
definitions, 

2  fa  ft)  ^  0.  (449) 


MAXIMUM  AND  MINIMUM  PROPERTIES.         137 


But  since  .  2  (cx  Px)  =  P 

and  2  <?i  =  1, 

2  fa  ft)  =  2  fa  P!  log  Px)  -  P  log  P.  (450) 

This  proves  (448),  and  shows  that  the  sign  =  will  hold  only 

when 

P1  =  P,    P2  =  P,    etc. 

for  all  phases,  i.  e.,  only  when  the  distribution  in  phase  of  the 
original  ensembles  are  all  identical. 

Theorem  IX.  A  uniform  distribution  of  a  given  number  of 
systems  within  given  limits  of  phase  gives  a  less  average  index 
of  probability  of  phase  than  any  other  distribution. 

Let  77  be  the  constant  index  of  the  uniform  distribution,  and 
T?  +  A?;  the  index  of  some  other  distribution.  Since  the  num- 
ber of  systems  within  the  given  limits  is  the  same  in  the  two 
distributions  we  have 

J.  .  .  Je"+  A*  dp,...  dqn  =  J.  .  .  J>  dp,  .  .  .  dqn,     (451) 

where  the  integrations,  like  those  which  follow,  are  to  be 
taken  within  the  given  limits.  The  proposition  to  be  proved 
may  be  written 


Pl...  dqn  >       .  .  .      ,;  JdPl  .  .  .  dqn,   (452) 

or,  since  77  is  constant, 

l  ...dqn  >.  .  .rjdp!  .  .  .  dqn.       (453) 


In  (451)  also  we  may  cancel  the  constant  factor  e^,  and  multiply 
by  the  constant  factor  (rj  +  1).     This  gives 


f.  .  . 


The  subtraction  of  this  equation  will  not  alter  the  inequality 
to  be  proved,  which  may  therefore  be  written 

/.  .  ./(A,  -  1)  /"  dPl...  dj.  >/.  .  ./-  cfc  .  .  .  dj. 


138          MAXIMUM  AND  MINIMUM  PROPERTIES. 

f . . .  f (AM  eA"  -  /"  +  1)  dPl  .  . .  dqn  >  0.          (454) 
J        J 


or 

Since  the  parenthesis  in  this  expression  represents  a  positive 
value,  except  when  it  vanishes  for  AT;  =  0,  the  integral  will 
be  positive  unless  AT?  vanishes  everywhere  within  the  limits, 
which  would  make  the  difference  of  the  two  distributions 
vanish.  The  theorem  is  therefore  proved. 


CHAPTER  XII. 

ON  THE  MOTION  OF  SYSTEMS  AND  ENSEMBLES  OF  SYS- 
TEMS THROUGH  LONG  PERIODS  OF  TIME. 

AN  important  question  which  suggests  itself  in  regard  to  any 
case  of  dynamical  motion  is  whether  the  system  considered 
will  return  in  the  course  of  time  to  its  initial  phase,  or,  if  it 
will  not  return  exactly  to  that  phase,  whether  it  will  do  so  to 
any  required  degree  of  approximation  in  the  course  of  a  suffi- 
ciently long  time.  To  be  able  to  give  even  a  partial  answer 
to  such  questions,  we  must  know  something  in  regard  to  the 
dynamical  nature  of  the  system.  In  the  following  theorem, 
the  only  assumption  in  this  respect  is  such  as  we  have  found 
necessary  for  the  existence  of  the  canonical  distribution. 

If  we  imagine  an  ensemble  of  identical  systems  to  be 
distributed  with  a  uniform  density  throughout  any  finite 
extension-in-phase,  the  number  of  the  systems  which  leave 
the  extension-in-phase  and  will  not  return  to  it  in  the  course 
of  time  is  less  than  any  assignable  fraction  of  the  whole 
number;  provided,  that  the  total  extension-in-phase  for  the 
systems  considered  between  two  limiting  values  of  the  energy 
is  finite,  these  limiting  values  being  less  and  greater  respec- 
tively than  any  of  the  energies  of  the  first-mentioned  exten- 
sion-in-phase. 

To  prove  this,  we  observe  that  at  the  moment  which  we 
call  initial  the  systems  occupy  the  given  extension-in-phase. 
It  is  evident  that  some  systems  must  leave  the  extension 
immediately,  unless  all  remain  in  it  forever.  Those  systems 
which  leave  the  extension  at  the  first  instant,  we  shall  call 
the  front  of  the  ensemble.  It  will  be  convenient  to  speak  of 
this  front  as  generating  the  extension-in-phase  through  which  it 
passes  in  the  course  of  time,  as  in  geometry  a  surface  is  said  to 


140  MOTION  OF  SYSTEMS  AND  ENSEMBLES 

generate  the  volume  through  which  it  passes.  In  equal  times 
the  front  generates  equal  extensions  in  phase.  This  is  an 
immediate  consequence  of  the  principle  of  conservation  of 
extension-in-phase^  unless  indeed  we  prefer  to  consider  it  as 
a  slight  variation  in  the  expression  of  that  principle.  For  in 
two  equal  short  intervals  of  time  let  the  extensions  generated 
be  A  and  B.  (We  make  the  intervals  short  simply  to  avoid 
the  complications  in  the  enunciation  or  interpretation  of  the 
principle  which  would  arise  when  the  same  extension-in-phase 
is  generated  more  than  once  in  the  interval  considered.)  Now 
if  we  imagine  that  at  a  given  instant  systems  are  distributed 
throughout  the  extension  A,  it  is  evident  that  the  same 
systems  will  after  a  certain  tune  occupy  the  extension  B, 
which  is  therefore  equal  to  A  in  virtue  of  the  principle  cited. 
The  front  of  the  ensemble,  therefore,  goes  on  generating 
equal  extensions  in  equal  times.  But  these  extensions  are 
included  in  a  finite  extension,  viz.,  that  bounded  by  certain 
limiting  values  of  the  energy.  Sooner  or  later,  therefore, 
the  front  must  generate  phases  which  it  has  before  generated. 
Such  second  generation  of  the  same  phases  must  commence 
with  the  initial  phases.  Therefore  a  portion  at  least  of  the 
front  must  return  to  the  original  extension-in-phase.  The 
same  is  of  course  true  of  the  portion  of  the  ensemble  which 
follows  that  portion  of  the  front  through  the  same  phases  at 
a  later  time. 

It  remains  to  consider  how  large  the  portion  of  the  ensemble 
is,  which  will  return  to  the  original  extension-in-phase.  There 
can  be  no  portion  of  the  given  extension-in-phase,  the  systems 
of  which  leave  the  extension  and  do  not  return.  For  we  can 
prove  for  any  portion  of  the  extension  as  for  the  whole,  that 
at  least  a  portion  of  the  systems  leaving  it  will  return. 

We  may  divide  the  given  extension-in-phase  into  parts  as 
follows.  There  may  be  parts  such  that  the  systems  within 
them  will  never  pass  out  of  them.  These  parts  may  indeed 
constitute  the  whole  of  the  given  extension.  But  if  the  given 
extension  is  very  small,  these  parts  will  in  general  be  non- 
existent. There  may  be  parts  such  that  systems  within  them 


THROUGH  LONG  PERIODS  OF  TIME.  141 

will  all  pass  out  of  the  given  extension  and  all  return  within 
it.  The  whole  of  the  given  extension-in-phase  is  made  up  of 
parts  of  these  two  kinds.  This  does  not  exclude  the  possi- 
bility of  phases  on  the  boundaries  of  such  parts,  such  that 
systems  starting  with  those  phases  would  leave  the  extension 
and  never  return.  But  in  the  supposed  distribution  of  an 
ensemble  of  systems  with  a  uniform  density-in-phase,  such 
systems  would  not  constitute  any  assignable  fraction  of  the 
whole  number. 

These  distinctions  may  be  illustrated  by  a  very  simple 
example.  If  we  consider  the  motion  of  a  rigid  body  of 
which  one  point  is  fixed,  and  which  is  subject  to  no  forces, 
we  find  three  cases.  (1)  The  motion  is  periodic.  (2)  The 
system  will  never  return  to  its  original  phase,  but  will  return 
infinitely  near  to  it.  (3)  The  system  will  never  return  either 
exactly  or  approximately  to  its  original  phase.  But  if  we 
consider  any  extension-in-phase,  however  small,  a  system 
leaving  that  extension  will  return  to  it  except  in  the  case 
called  by  Poinsot  *  singular,'  viz.,  when  the  motion  is  a 
rotation  about  an  axis  lying  in  one  of  two  planes  having 
a  fixed  position  relative  to  the  rigid  body.  But  all  such 
phases  do  not  constitute  any  true  extension-in-phase  in  the 
sense  in  which  we  have  defined  and  used  the  term.* 

In  the  same  way  it  may  be  proved  that  the  systems  in  a 
canonical  ensemble  which  at  a  given  instant  are  contained 
within  any  finite  extension-in-phase  will  in  general  return  to 

*  An  ensemble  of  systems  distributed  in  phase  is  a  less  simple  and  ele- 
mentary conception  than  a  single  system.  But  by  the  consideration  of 
suitable  ensembles  instead  of  single  systems,  we  may  get  rid  of  the  incon- 
venience of  having  to  consider  exceptions  formed  by  particular  cases  of  the 
integral  equations  of  motion,  these  cases  simply  disappearing  when  the 
ensemble  is  substituted  for  the  single  system  as  a  subject  of  study.  This 
is  especially  true  when  the  ensemble  is  distributed,  as  in  the  case  called 
canonical,  throughout  an  extension-in-phase.  In  a  less  degree  it  is  true  of 
the  microcanonical  ensemble,  which  does  not  occupy  any  extension-in-phase, 
(in  the  sense  in  which  we  have  used  the  term,)  although  it  is  convenient  to 
regard  it  as  a  limiting  case  with  respect  to  ensembles  which  do,  as  we  thus 
gain  for  the  subject  some  part  of  the  analytical  simplicity  which  belongs  to 
the  theory  of  ensembles  which  occupy  true  extensions-in-phase. 


142  MOTION  OF  SYSTEMS  AND  ENSEMBLES 

that  extension-in-phase,  if  they  leave  it,  the  exceptions,  i.  g., 
the  number  which  pass  out  of  the  extension-in-phase  and  do 
not  return  to  it,  being  less  than  any  assignable  fraction  of  the 
whole  number.  In  other  words,  the  probability  that  a  system 
taken  at  random  from  the  part  of  a  canonical  ensemble  which 
is  contained  within  any  given  extension-in-phase,  will  pass  out 
of  that  extension  and  not  return  to  it,  is  zero. 

A  similar  theorem  may  be  enunciated  with  respect  to  a 
roicrocanonical  ensemble.  Let  us  consider  the  fractional  part 
of  such  an  ensemble  which  lies  within  any  given  limits  of 
phase.  This  fraction  we  shall  denote  by  F.  It  is  evidently 
constant  in  time  since  the  ensemble  is  in  statistical  equi- 
librium. The  systems  within  the  limits  will  not  in  general 
remain  the  same,  but  some  will  pass  out  in  each  unit  of  time 
while  an  equal  number  come  in.  Some  may  pass  out  never 
to  return  within  the  limits.  But  the  number  which  in  any 
time  however  long  pass  out  of  the  limits  never  to  return  will 
not  bear  any  finite  ratio  to  the  number  within  the  limits  at 
a  given  instant.  For,  if  it  were  otherwise,  let  /  denote  the 
fraction  representing  such  ratio  for  the  tune  T.  Then,  in 
the  time  T,  the  number  which  pass  out  never  to  return  will 
bear  the  ratio  f  F  to  the  whole  number  in  the  ensemble,  and 
in  a  time  exceeding  T/(fF)  the  number  which  pass  out  of 
the  limits  never  to  return  would  exceed  the  total  number 
of  systems  in  the  ensemble.  The  proposition  is  therefore 
proved. 

This  proof  will  apply  to  the  cases  before  considered,  and 
may  be  regarded  as  more  simple  than  that  which  was  given. 
It  may  also  be  applied  to  any  true  case  of  statistical  equilib- 
rium. By  a  true  case  of  statistical  equilibrium  is  meant  such 
as  may  be  described  by  giving  the  general  value  of  the  prob- 
ability that  an  unspecified  system  of  the  ensemble  is  con- 
tained within  any  given  limits  of  phase.* 

*  An  ensemble  in  which  the  systems  are  material  points  constrained  to 
move  in  vertical  circles,  with  just  enough  energy  to  carry  them  to  the 
highest  points,  cannot  afford  a  true  example  of  statistical  equilibrium.  For 
any  other  value  of  the  energy  than  the  critical  value  mentioned,  we  might 


THROUGH  LONG  PERIODS  OF  TIME.  143 

Let  us  next  consider  whether  an  ensemble  of  isolated 
systems  has  any  tendency  in  the  course  of  time  toward  a 
state  of  statistical  equilibrium. 

There  are  certain  functions  of  phase  which  are  constant  in 
time.  The  distribution  of  the  ensemble  with  respect  to  the 
values  of  these  functions  is  necessarily  invariable,  that  is, 
the  number  of  systems  within  any  limits  which  can  be 
specified  in  terms  of  these  functions  cannot  vary  in  the  course 
of  time.  The  distribution  in  phase  which  without  violating 
this  condition  gives  the  least  value  of  the  average  index  of 
probability  of  phase  (77)  is  unique,  and  is  that  in  which  the 

in  various  ways  describe  an  ensemble  in  statistical  equilibrium,  while  the 
same  language  applied  to  the  critical  value  of  the  energy  would  fail  to  do 
so.  Thus,  if  we  should  say  that  the  ensemble  is  so  distributed  that  the 
probability  that  a  system  is  in  any  given  part  of  the  circle  is  proportioned 
to  the  time  which  a  single  system  spends  in  that  part,  motion  in  either  direc- 
tion being  equally  probable,  we  should  perfectly  define  a  distribution  in  sta- 
tistical equilibrium  for  any  value  of  the  energy  except  the  critical  value 
mentioned  above,  but  for  this  value  of  the  energy  all  the  probabilities  in 
question  would  vanish  unless  the  highest  point  is  included  in  the  part  of  the 
circle  considered,  in  which  case  the  probability  is  unity,  or  forms  one  of  its 
limits,  in  which  case  the  probability  is  indeterminate.  Compare  the  foot-note 
on  page  118. 

A  still  more  simple  example  is  afforded  by  the  uniform  motion  of  a 
material  point  in  a  straight  line.  Here  the  impossibility  of  statistical  equi- 
librium is  not  limited  to  any  particular  energy,  and  the  canonical  distribu- 
tion as  well  as  the  microcanonical  is  impossible. 

These  examples  are  mentioned  here  in  order  to  show  the  necessity  of 
caution  in  the  application  of  the  above  principle,  with  respect  to  the  question 
whether  we  have  to  do  with  a  true  case  of  statistical  equilibrium. 

Another  point  in  respect  to  which  caution  must  be  exercised  is  that  the 
part  of  an  ensemble  of  which  the  theorem  of  the  return  of  systems  is  asserted 
should  be  entirely  denned  by  limits  within  which  it  is  contained,  and  not  by 
any  such  condition  as  that  a  certain  function  of  phase  shall  have  a  given 
value.  This  is  necessary  in  order  that  the  part  of  the  ensemble  which  is 
considered  should  be  any  assignable  fraction  of  the  whole.  Thus,  if  we  have 
a  canonical  ensemble  consisting  of  material  points  in  vertical  circles,  the 
theorem  of  the  return  of  systems  may  be  applied  to  a  part  of  the  ensemble 
defined  as  cqntained  in  a  given  part  of  the  circle.  But  it  may  not  be  applied 
in  all  cases  to  a  part  of  the  ensemble  defined  as  contained  in  a  given  part 
of  the  circle  and  having  a  given  energy.  It  would,  in  fact,  express  the  exact 
opposite  of  the  truth  when  the  given  energy  is  the  critical  value  mentioned 
above. 


144  MOTION  OF  SYSTEMS  AND  ENSEMBLES 

index  of  probability  (77)  is  a  function  of  the  functions  men- 
tioned.* It  is  therefore  a  permanent  distribution,  f  and  the 
only  permanent  distribution  consistent  with  the  invariability 
of  the  distribution  with  respect  to  the  functions  of  phase 
which  are  constant  in  time. 

It  would  seem,  therefore,  that  we  might  find  a  sort  of  meas- 
ure of  the  deviation  of  an  ensemble  from  statistical  equilibrium 
in  the  excess  of  the  average  index  above  the  minimum  which  is 
consistent  with  the  condition  of  the  invariability  of  the  distri- 
bution with  respect  to  the  constant  functions  of  phase.  But 
we  have  seen  that  the  index  of  probability  is  constant  in  time 
for  each  system  of  the  ensemble.  The  average  index  is  there- 
fore constant,  and  we  find  by  this  method  no  approach  toward 
statistical  equilibrium  in  the  course  of  time. 

Yet  we  must  here  exercise  great  caution.  One  function 
may  approach  indefinitely  near  to  another  function,  while 
some  quantity  determined  by  the  first  does  not  approach  the 
corresponding  quantity  determined  by  the  second.  A  line 
joining  two  points  may  approach  indefinitely  near  to  the 
straight  line  joining  them,  while  its  length  remains  constant. 
We  may  find  a  closer  analogy  with  the  case  under  considera- 
tion in  the  effect  of  stirring  an  incompressible  liquid.^  In 
space  of  2  n  dimensions  the  case  might  be  made  analyti- 
cally identical  with  that  of  an  ensemble  of  systems  of  n 
degrees  of  freedom,  but  the  analogy  is  perfect  in  ordinary- 
space.  Let  us  suppose  the  liquid  to  contain  a  certain  amount 
of  coloring  matter  which  does  not  affect  its  hydrodynamic 
properties.  Now  the  state  in  which  the  density  of  the  coloring 
matter  is  uniform,  i.  e.,  the  statt,  of  perfect  mixture,  which  is 
a  sort  of  state  of  equilibrium  in  this  respect  that  the  distribu- 
tion of  the  coloring  matter  in  space  is  not  affected  by  the 
internal  motions  of  the  liquid,  is  characterized  by  a  minimum 

*  See  Chapter  XI,  Theorem  IV. 

t  See  Chapter  IV,  sub  init. 

J  By  liquid  is  here  meant  the  continuous  body  of  theoretical  hydrody- 
namics, and  not  anything  of  the  molecular  structure  and  molecular  motions 
of  real  liquids. 


THROUGH  LONG  PERIODS   OF  TIME.  145 

value  of  the  average  square  of  the  density  of  the  coloring 
matter.  Let  us  suppose,  however,  that  the  coloring  matter  is 
distributed  with  a  variable  density.  If  we  give  the  liquid  any 
motion  whatever,  subject  only  to  the  hydrodynamic  law  of 
incompressibility,  —  it  may  be  a  steady  flux,  or  it  may  vary 
with  the  time, — the  density  of  the  coloring  matter  at  any 
same  point  of  the  liquid  will  be  unchanged,  and  the  average 
square  of  this  density  will  therefore  be  unchanged.  Yet  no 
fact  is  more  familiar  to  us  than  that  stirring  tends  to  bring  a 
liquid  to  a  state  of  uniform  mixture,  or  uniform  densities  of 
its  components,  which  is  characterized  by  minimum  values 
of  the  average  squares  of  these  densities.  It  is  quite  true  that 
in  the  physical  experiment  the  result  is  hastened  by  the 
process  of  diffusion,  but  the  result  is  evidently  not  dependent 
on  that  process. 

The  contradiction  is  to  be  traced  to  the  notion  of  the  density 
of  the  coloring  matter,  and  the  process  by  which  this  quantity 
is  evaluated.  This  quantity  is  the  limiting  ratio  of  the 
quantity  of  the  coloring  matter  in  an  element  of  space  to  the 
volume  of  that  element.  Now  if  we  should  take  for  our  ele- 
ments of  volume,  after  any  amount  of  stirring,  the  spaces 
occupied  by  the  same  portions  of  the  liquid  which  originally 
occupied  any  given  system  of  elements  of  volume,  the  densi- 
ties of  the  coloring  matter,  thus  estimated,  would  be  identical 
with  the  original  densities  as  determined  by  the  given  system 
of  elements  of  volume.  Moreover,  if  at  the  end  of  any  finite 
amount  of  stirring  we  should  take  our  elements  of  volume  in 
any  ordinary  form  but  sufficiently  small,  the.  average  square 
of  the  density  of  the  coloring  matter,  as  determined  by  such 
element  of  volume,  would  approximate  to  any  required  degree 
to  its  value  before  the  stirring.  But  if  we  take  any  element 
of  space  of  fixed  position  and  dimensions,  we  may  continue 
the  stirring  so  long  that  the  densities  of  the  colored  liquid 
estimated  for  these  fixed  elements  will  approach  a  uniform 
limit,  viz.',  that  of  perfect  mixture. 

The  case  is  evidently  one  of  those  in  which  the  limit  of  a 
limit  has  different  values,  according  to  the  order  in  which  we 

10 


146  MOTION  OF  SYSTEMS  AND  ENSEMBLES 

apply  the  processes  of  taking  a  limit.  If  treating  the  elements 
of  volume  as  constant,  we  continue  the  stirring  indefinitely, 
we  get  a  uniform  density,  a  result  not  affected  by  making  the 
elements  as  small  as  we  choose ;  but  if  treating  the  amount  of 
stirring  as  finite,  we  diminish  indefinitely  the  elements  of 
volume,  we  get  exactly  the  same  distribution  in  density  as 
before  the  stirring,  a  result  which  is  not  affected  by  con- 
tinuing the  stirring  as  long  as  we  choose.  The  question  is 
largely  one  of  language  and  definition.  One  may  perhaps  be 
allowed  to  say  that  a  finite  amount  of  stirring  will  not  affect 
the  mean  square  of  the  density  of  the  coloring  matter,  but  an 
infinite  amount  of  stirring  may  be  regarded  as  producing  a 
condition  in  which  the  mean  square  of  the  density  has  its 
minimum  value,  and  the  density  is  uniform.  We  may  cer- 
tainly say  that  a  sensibly  uniform  density  of  the  colored  com- 
ponent may  be  produced  by  stirring.  Whether  the  time 
required  for  this  result  would  be  long  or  short  depends  upon 
the  nature  of  the  motion  given  to  the  liquid,  and  the  fineness 
of  our  method  of  evaluating  the  density. 

All  this  may  appear  more  distinctly  if  we  consider  a  special 
case  of  liquid  motion.  Let  us  imagine  a  cylindrical  mass  of 
liquid  of  which  one  sector  of  90°  is  black  and  the  rest  white. 
Let  it  have  a  motion  of  rotation  about  the  axis  of  the  cylinder 
in  which  the  angular  velocity  is  a  function  of  the  distance 
from  the  axis.  In  the  course  of  time  the  black  and  the  white 
parts  would  become  drawn  out  into  thin  ribbons,  which  would 
be  wound  spirally  about  the  axis.  The  thickness  of  these  rib- 
bons would  diminish  without  limit,  and  the  liquid  would  there- 
fore tend  toward  a  state  of  perfect  mixture  of  the  black  and 
white  portions.  That  is,  in  any  given  element  of  space,  the 
proportion  of  the  black  and  white  would  approach  1 :  3  as  a  limit. 
Yet  after  any  finite  time,  the  total  volume  would  be  divided 
into  two  parts,  one  of  which  would  consist  of  the  white  liquid 
exclusively,  and  the  other  of  the  black  exclusively.  If  the 
coloring  matter,  instead  of  being  distributed  initially  with  a 
uniform  density  throughout  a  section  of  the  cylinder,  were 
distributed  with  a  density  represented  by  any  arbitrary  func- 


THROUGH  LONG  PERIODS   OF  TIME.  147 

tion  of  the  cylindrical  coordinates  r,  6  and  2,  the  effect  of  the 
same  motion  continued  indefinitely  would  be  an  approach  to 
a  condition  in  which  the  density  is  a  function  of  r  and  z  alone. 
In  this  limiting  condition,  the  average  square  of  the  density 
would  be  less  than  in  the  original  condition,  when  the  density 
was  supposed  to  vary  with  0,  although  after  any  finite  time 
the  average  square  of  the  density  would  be  the  same  as  at 
first. 

If  we  limit  our  attention  to  the  motion  in  a  single  plane 
perpendicular  to  the  axis  of  the  cylinder,  we  have  something 
which  is  almost  identical  with  a  diagrammatic  representation 
of  the  changes  in  distribution  in  phase  of  an  ensemble  of 
systems  of  one  degree  of  freedom,  in  which  the  motion  is 
periodic,  the  period  varying  with  the  energy,  as  in  the  case  of 
a  pendulum  swinging  in  a  circular  arc.  If  the  coordinates 
and  momenta  of  the  systems  are  represented  by  rectangu- 
lar coordinates  in  the  diagram,  the  points  in  the  diagram 
representing  the  changing  phases  of  moving  systems,  will 
move  about  the  origin  in  closed  curves  of  constant  energy. 
The  motion  will  be  such  that  areas  bounded  by  points  repre- 
senting moving  systems  will  be  preserved.  The  only  differ- 
ence between  the  motion  of  the  liquid  and  the  motion  in  the 
diagram  is  that  in  one  case  the  paths  are  circular,  and  in  the 
other  they  differ  more  or  less  from  that  form. 

When  the  energy  is  proportional  to  p2  +  q2  the  curves  of 
constant  energy  are  circles,  and  the  period  is  independent  of 
the  energy.  There  is  then  no  tendency  toward  a  state  of  sta- 
tistical equilibrium.  The  diagram  turns  about  the  origin  with- 
out change  of  form.  This  corresponds  to  the  case  of  liquid 
motion,  when  the  liquid  revolves  with  a  uniform  angular 
velocity  like  a  rigid  solid. 

The  analogy  between  the  motion  of  an  ensemble  of  systems 
in  an  extension-in-phase  and  a  steady  current  in  an  incompres- 
sible liquid,  and  the  diagrammatic  representation  of  the  case 
of  one  degree  of  freedom,  which  appeals  to  our  geometrical  in- 
tuitions, may  be  sufficient  to  show  how  the  conservation  of 
density  in  phase,  which  involves  the  conservation  of  the 


148  MOTION  OF  SYSTEMS  AND  ENSEMBLES 

average  value  of  the  index  of  probability  of  phase,  is  consist- 
ent with  an  approach  to  a  limiting  condition  in  which  that 
average  value  is  less.  We  might  perhaps  fairly  infer  from 
such  considerations  as  have  been  adduced  that  an  approach 
to  a  limiting  condition  of  statistical  equilibrium  is  the  general 
rule,  when  the  initial  condition  is  not  of  that  character.  But 
the  subject  is  of  such  importance  that  it  seems  desirable  to 
give  it  farther  consideration. 

Let  us  suppose  that  the  total  extension-in-phase  for  the 
kind  of  system  considered  to  be  divided  into  equal  elements 
(D  V)  which  are  very  small  but  not  infinitely  small.  Let  us 
imagine  an  ensemble  of  systems  distributed  in  this  extension 
in  a  manner  represented  by  the  index  of  probability  77,  which 
is  an  arbitrary  function  of  the  phase  subject  only  to  the  re- 
striction expressed  by  equation  (46)  of  Chapter  I.  We  shall 
suppose  the  elements  D  V  to  be  so  small  that  rj  may  in  gen- 
eral be  regarded  as  sensibly  constant  within  any  one  of  them 
at  the  initial  moment.  Let  the  path  of  a  system  be  defined  as 
the  series  of  phases  through  which  it  passes. 

At  the  initial  moment  (£')  a  certain  system  is  in  an  element 
of  extension  DVf.  Subsequently,  at  the  time  £",  the  same 
system  is  in  the  element  DV".  Other  systems  which  were 
at  first  in  DV  will  at  the  time  t"  be  in  DV",  but  not  all, 
probably.  The  systems  which  were  at  first  in  DV1  will  at 
the  time  t'f  occupy  an  extension-in-phase  exactly  as  large  as  at 
first.  But  it  will  probably  be  distributed  among  a  very  great 
number  of  the  elements  (DV)  into  which  we  have  divided 
the  total  extension-in-phase.  If  it  is  not  so,  we  can  generally 
take  a  later  time  at  which  it  will  be  so.  There  will  be  excep- 
tions to  this  for  particular  laws  of  motion,  but  we  will  con- 
fine ourselves  to  what  may  fairly  be  called  the  general  case. 
Only  a  very  small  part  of  the  systems  initially  in  D  V  will 
be  found  in  DV"  at  the  time  t",  and  those  which  are  found  in 
DV"  at  that  time  were  at  the  initial  moment  distributed 
among  a  very  large  number  of  elements  D  V. 

What  is  important  for  our  purpose  is  the  value  of  77,  the 
index  of  probability  of  phase  in  the  element  DV"  at  the  time 


THROUGH  LONG  PERIODS   OF  TIME.  149 

t".     In  the  part  of  DV"  occupied  by  systems  which  at  the 
time  if  were  in  DV  the  value  of  77  will  be  the  same  as  its 
value  in  D  V  at  the  time  tr,  which  we  shall  call  77'.     In  the 
parts  of  DV"  occupied  by  systems  which   at  if  were  in  ele- 
ments very  near  to  D  V  we  may  suppose  the  value  of  77  to 
vary  little  from  T?'.     We  cannot  assume  this  in  regard  to  parts 
of   DV"  occupied  by  systems  which  at  tf  were  in  elements 
remote  from  DV.      We  want,  therefore,  some  idea  of  the 
nature  of  the  extension-in-phase  occupied  at  tf  by  the  sys- 
tems which  at  t"  will  occupy  D  V".      Analytically,  the  prob- 
lem is  identical  with  finding  the  extension  occupied  at   t" 
by  the  systems  which  at  t1  occupied  DV.     Now  the  systems 
in  D  V"  which  lie  on  the  same  path  as  the  system  first  con- 
sidered, evidently  arrived  at  DV"  at  nearly  the  same  time, 
and  must  have  left  D  V1  at  nearly  the  same  time,  and  there- 
fore at  if  were  in  or  near  DV.     We  may  therefore  take  T/  as 
the  value  for  these  systems.     The  same  essentially  is  true  of 
systems  in  DV"  which  he  on  paths  very  close  to  the  path 
already  considered.     But  with  respect  to  paths  passing  through 
D  V  and  D  V",  but  not  so  close  to  the  first  path,  we  cannot 
assume  that  the  time  required  to  pass  from  DV  to  D V"  is 
nearly  the  same  as  for  the  first  path.     The  difference  of  the 
times  required  may  be  small  in  comparison  with  £"-£',  but  as 
this  interval  can  be  as  large  as  we  choose,  the  difference  of  the 
times  required  in  the  different  paths  has  no  limit  to  its  pos- 
sible value.     Now  if  the  case  were  one  of  statistical  equilib- 
rium, the  value  of  77  would  be  constant  in  any  path,  and  if  all 
the  paths  which  pass  through  DV1  also  pass  through  or  near 
D  V,  the  value  of  77  throughout  D  V"  will  vary  little  from 
?;'.     But  when  the  case  is  not  one  of  statistical  equilibrium, 
we  cannot  draw  any  such  conclusion.     The  only  conclusion 
which  we  can  draw  with  respect  to  the  phase  at  t1  of  the  sys- 
tems which  at  t"  are  in  DV"  is  that  they  are  nearly  on  the 
same  patji. 

Now  if  we  should  make  a  new  estimate  of  indices  of  prob- 
ability of  phase  at  the  time  t",  using  for  this  purpose  the 
elements  D  V,  —  that  is,  if  we  should  divide  the  number  of 


150  MOTION  OF  SYSTEMS  AND  ENSEMBLES 

systems  in  JDF",  for  example,  by  the  total  number  of  systems, 
and  also  by  the  extension-in-phase  of  the  element,  and  take 
the  logarithm  of  the  quotient,  we  would  get  a  number  which 
would  be  less  than  the  average  value  of  rj  for  the  systems 
within  D  V"  based  on  the  distribution  in  phase  at  the  time  t1.* 
Hence  the  average  value  of  77  for  the  whole  ensemble  of 
systems  based  on  the  distribution  at  t"  will  be  less  than  the 
average  value  based  on  the  distribution  at  t'. 

We  must  not  forget  that  there  are  exceptions  to  this  gen- 
eral rule.  These  exceptions  are  in  cases  in  which  the  laws 
of  motion  are  such  that  systems  having  small  differences 
of  phase  will  continue  always  to  have  small  differences  of 
phase. 

It  is  to  be  observed  that  if  the  average  index  of  probability  in 
an  ensemble  may  be  said  in  some  sense  to  have  a  less  value  at 
one  tune  than  at  another,  it  is  not  necessarily  priority  in  tune 
which  determines  the  greater  average  index.  If  a  distribution, 
which  is  not  one  of  statistical  equilibrium,  should  be  given 
for  a  time  £',  and  the  distribution  at  an  earlier  time  t"  should 
be  defined  as  that  given  by  the  corresponding  phases,  if  we 
increase  the  interval  leaving  t'  fixed  and  taking  ttf  at  an  earlier 
and  earlier  date,  the  distribution  at  t"  will  in  general  approach 
a  limiting  distribution  which  is  in  statistical  equilibrium.  The 
determining  difference  in  such  cases  is  that  between  a  definite 
distribution  at  a  definite  time  and  the  limit  of  a  varying  dis- 
tribution when  the  moment  considered  is  carried  either  forward 
or  backward  indefinitely,  f 

But  while  the  distinction  of  prior  and  subsequent  events 
may  be  immaterial  with  respect  to  mathematical  fictions,  it  is 
quite  otherwise  with  respect  to  the  events  of  the  real  world. 
It  should  not  be  forgotten,  when  our  ensembles  are  chosen  to 
illustrate  the  probabilities  of  events  in  the  real  world,  that 

*  See  Chapter  XI,  Theorem  IX. 

t  One  may  compare  the  kinematical  truism  that  when  two  points  are 
moving  with  uniform  velocities,  (with  the  single  exception  of  the  case  where 
the  relative  motion  is  zero,)  their  mutual  distance  at  any  definite  time  is  less 
than  f  or  t  =  <x> ,  or  t  =  —  oo . 


THROUGH  LONG  PERIODS   OF  TIME.  151 

while  the  probabilities  of  subsequent  events  may  often  be 
determined  from  the  probabilities  of  prior  events,  it  is  rarely 
the  case  that  probabilities  of  prior  events  can  be  determined  \j 
from  those  of  subsequent  events,  for  we  are  rarely  justified  in 
excluding  the  consideration  of  the  antecedent  probability  of 
the  prior  events. 

It  is  worthy  of  notice  that  to  take  a  system  at  random  from 
an  ensemble  at  a  date  chosen  at  random  from  several  given 
dates,  t',  t",  etc.,  is  practically  the  same  thing  as  to  take  a  sys- 
tem at  random  from  the  ensemble  composed  of  all  the  systems 
of  the  given  ensemble  in  their  phases  at  the  time  £',  together 
with  the  same  systems  in  their  phases  at  the  time  t/;,  etc.  By 
Theorem  VIII  of  Chapter  XI  this  will  give  an  ensemble  in 
which  the  average  index  of  probability  will  be  less  than  in 
the  given  ensemble,  except  in  the  case  when  the  distribution 
in  the  given  ensemble  is  the  same  at  the  times  tr,  t'f,  etc. 
Consequently,  any  indefiniteness  in  the  time  in  which  we  take 
a  system  at  random  from  an  ensemble  has  the  practical  effect 
of  diminishing  the  average  index  of  the  ensemble  from  which 
the  system  may  be  supposed  to  be  drawn,  except  when  the 
given  ensemble  is  in  statistical  equilibrium. 


CHAPTER  XIII. 

EFFECT    OF   VARIOUS    PROCESSES    ON  AN  ENSEMBLE  OF 

SYSTEMS. 

IN  the  last  chapter  and  in  Chapter  I  we  have  considered  the 
changes  which  take  place  in  the  course  of  time  in  an  ensemble 
of  isolated  systems.  Let  us  now  proceed  to  consider  the 
changes  which  will  take  place  in  an  ensemble  of  systems  under 
external  influences.  These  external  influences  will  be  of  two 
kinds,  the  variation  of  the  coordinates  which  we  have  called 
external,  and  the  action  of  other  ensembles  of  systems.  The 
essential  difference  of  the  two  kinds  of  influence  consists  in 
this,  that  the  bodies  to  which  the  external  coordinates  relate 
are  not  distributed  in  phase,  while  in  the  case  of  interaction 
of  the  systems  of  two  ensembles,  we  have  to  regard  the  fact 
that  both  are  distributed  in  phase.  To  find  the  effect  pro- 
duced on  the  ensemble  with  which  we  are  principally  con- 
cerned, we  have  therefore  to  consider  single  values  of  what 
we  have  called  external  coordinates,  but  an  infinity  of  values 
of  the  internal  coordinates  of  any  other  ensemble  with  which 
there  is  interaction. 

Or,  —  to  regard  the  subject  from  another  point  of  view,  — 
the  action  between  an  unspecified  system  of  an  ensemble  and 
the  bodies  represented  by  the  external  coordinates,  is  the 
action  between  a  system  imperfectly  determined  with  respect 
to  phase  and  one  which  is  perfectly  determined ;  while  the 
interaction  between  two  unspecified  systems  belonging  to 
different  ensembles  is  the  action  between  two  systems  both  of 
which  are  imperfectly  determined  with  respect  to  phase.* 

We  shall  suppose  the  ensembles  which  we  consider  to  be 
distributed  in  phase  in  the  manner  described  in  Chapter  I,  and 

*  In  the  development  of  the  subject,  we  shall  find  that  this  distinction 
corresponds  to  the  distinction  in  thermodynamics  between  mechanical  and 
thermal  action. 


EFFECT  OF  VARIOUS  PROCESSES.  153 

represented  by  the  notations  of  that  chapter,  especially  by  the 
index  of  probability  of  phase  (??).  There  are  therefore  2  n 
independent  variations  in  the  phases  which  constitute  the 
ensembles  considered.  This  excludes  ensembles  like  the 
microcanonical,  in  which,  as  energy  is  constant,  there  are 
only  2  n  —  1  independent  variations  of  phase.  This  seems 
necessary  for  the  purposes  of  a  general  discussion.  For 
although  we  may  imagine  a  microcanonical  ensemble  to  have 
a  permanent  existence  when  isolated  from  external  influences, 
the  effect  of  such  influences  would  generally  be  to  destroy  the 
uniformity  of  energy  in  the  ensemble.  Moreover,  since  the 
microcanonical  ensemble  may  be  regarded  as  a  limiting  case  of 
such  ensembles  as  are  described  in  Chapter  I,  (and  that  in 
more  than  one  way,  as  shown  in  Chapter  X,)  the  exclusion  is 
rather  formal  than  real,  since  any  properties  which  belong  to 
the  microcanonical  ensemble  could  easily  be  derived  from  those 
of  the  ensembles  of  Chapter  I,  which  in  a  certain  sense  may 
be  regarded  as  representing  the  general  case. 

Let  us  first  consider  the  effect  of  variation  of  the  external 
coordinates.  We  have  already  had  occasion  to  regard  these 
quantities  as  variable  in  the  differentiation  of  certain  equations 
relating  to  ensembles  distributed  according  to  certain  laws 
called  canonical  or  microcanonical.  That  variation  of  the 
external  coordinates  was,  however,  only  carrying  the  atten- 
tion of  the  mind  from  an  ensemble  with  certain  values  of  the 
external  coordinates,  and  distributed  in  phase  according  to 
some  general  law  depending  upon  those  values,  to  another 
ensemble  with  different  values  of  the  external  coordinates,  and 
with  the  distribution  changed  to  conform  to  these  new  values. 

What  we  have  now  to  consider  is  the  effect  which  would 
actually  result  in  the  course  of  time  in  an  ensemble  of  systems 
in  which  the  external  coordinates  should  be  varied  in  any 
arbitrary  manner.  Let  us  suppose,  in  the  first  place,  that 
these  coordinates  are  varied  abruptly  at  a  given  instant,  being 
constant  both  before  and  after  that  instant.  By  the  definition 
of  the  external  coordinates  it  appears  that  this  variation  does 
not  affect  the  phase  of  any  system  of  the  ensemble  at  the  time 


154  EFFECT  OF  VARIOUS  PROCESSES 

when  it  takes  place.  Therefore  it  does  not  affect  the  index  of 
probability  of  phase  (77)  of  any  system,  or  the  average  value 
of  the  index  (?/)'  at  that  time.  And  if  these  quantities  are 
constant  in  time  before  the  variation  of  the  external  coordi- 
nates, and  after  that  variation,  their  constancy  hi  time  is  not 
interrupted  by  that  variation.  In  fact,  in  the  demonstration 
of  the  conservation  of  probability  of  phase  in  Chapter  I,  the 
variation  of  the  external  coordinates  was  not  excluded. 

But  a  variation  of  the  external  coordinates  will  in  general 
disturb  a  previously  existing  state  of  statistical  equilibrium. 
For,  although  it  does  not  affect  (at  the  first  instant)  the 
distribution-in-phase,  it  does  affect  the  condition  necessary  for 
equilibrium.  This  condition,  as  we  have  seen  in  Chapter  IV, 
is  that  the  index  of  probability  of  phase  shall  be  a  function  of 
phase  which  is  constant  in  time  for  moving  systems.  Now 
a  change  in  the  external  coordinates,  by  changing  the  forces 
which  act  on  the  systems,  will  change  the  nature  of  the 
functions  of  phase  which  are  constant  in  time.  Therefore, 
the  distribution  in  phase  which  was  one  of  statistical  equi- 
librium for  the  old  values  of  the  external  coordinates,  will  not 
be  such  for  the  new  values. 

Now  we  have  seen,  in  the  last  chapter,  that  when  the  dis- 
tribution-in-phase is  not  one  of  statistical  equilibrium,  an 
ensemble  of  systems  may,  and  in  general  will,  after  a  longer  or 
shorter  time,  come  to  a  state  which  may  be  regarded,  if  very 
small  differences  of  phase  are  neglected,  as  one  of  statistical 
equilibrium,  and  in  which  consequently  the  average  value  of 
the  index  (?;)  is  less  than  at  first.  It  is  evident,  therefore, 
that  a  variation  of  the  external  coordinates,  by  disturbing  a 
state  of  statistical  equilibrium,  may  indirectly  cause  a  diminu- 
tion, (in  a  certain  sense  at  least,)  of  the  value  of  rj. 

But  if  the  change  in  the  external  coordinates  is  very  small, 
the  change  in  the  distribution  necessary  for  equilibrium  will 
in  general  be  correspondingly  small.  Hence,  the  original  dis- 
tribution in  phase,  since  it  differs  little  from  one  which  would 
be  in  statistical  equilibrium  with  the  new  values  of  the  ex- 
ternal coordinates,  may  be  supposed  to  have  a  value  of  v 


ON  AN  ENSEMBLE   OF  SYSTEMS.  155 

which  differs  by  a  small  quantity  of  the  second  order  from 
the  minimum  value  which  characterizes  the  state  of  statistical 
equilibrium.  And  the  diminution  in  the  average  index  result- 
ing in  the  course  of  time  from  the  very  small  change  in  the 
external  coordinates,  cannot  exceed  this  small  quantity  of 
the  second  order. 

Hence  also,  if  the  change  in  the  external  coordinates  of  an 
ensemble  initially  in  statistical  equilibrium  consists  in  suc- 
cessive very  small  changes  separated  by  very  long  intervals  of 
time  in  which  the  disturbance  of  statistical  equilibrium  be- 
comes sensibly  effaced,  the  final  diminution  in  the  average 
index  of  probability  will  in  general  be  negligible,  although  the 
total  change  in  the  external  coordinates  is  large.  The  result 
will  be  the  same  if  the  change  in  the  external  coordinates 
takes  place  continuously  but  sufficiently  slowly. 

Even  in  cases  in  which  there  is  no  tendency  toward  the 
restoration  of  statistical  equilibrium  in  the  lapse  of  time,  a  varia- 
tion of  external  coordinates  which  would  cause,  if  it  took 
place  in  a  short  time,  a  great  disturbance  of  a  previous  state 
of  equilibrium,  may,  if  sufficiently  distributed  in  time,  produce 
no  sensible  disturbance  of  the  statistical  equilibrium. 

Thus,  in  the  case  of  three  degrees  of  freedom,  let  the  systems 
be  heavy  points  suspended  by  elastic  massless  cords,  and  let  the 
ensemble  be  distributed  in  phase  with  a  density  proportioned 
to  some  function  of  the  energy,  and  therefore  in  statistical  equi- 
librium. For  a  change  in  the  external  coordinates,  we  may 
take  a  horizontal  motion  of  the  point  of  suspension.  If  this 
is  moved  a  given  distance,  the  resulting  disturbance  of  the 
statistical  equilibrium  may  evidently  be  diminished  indefi- 
nitely by  diminishing  the  velocity  of  the  point  of  suspension. 
This  will  be  true  if  the  law  of  elasticity  of  the  string  is  such 
that  the  period  of  vibration  is  independent  of  the  energy,  in 
which  case  there  is  no  tendency  in  the  course  of  time  toward 
a  state  of  statistical  equilibrium,  as  well  as  in  the  more  general 
case,  in  which  there  is  a  tendency  toward  statistical  equilibrium. 

That  something  of  this  kind  will  be  true  in  general,  the 
following  considerations  will  tend  to  show. 


156  EFFECT  OF  VARIOUS  PROCESSES 

We  define  a  path  as  the  series  of  phases  through  which  a 
system  passes  in  the  course  of  time  when  the  external  co- 
ordinates have  fixed  values.  When  the  external  coordinates 
are  varied,  paths  are  changed.  The  path  of  a  phase  is  the 
path  to  which  that  phase  belongs.  With  reference  to  any 
ensemble  of  systems  we  shall  denote  by  27|p  the  average  value 
of  the  density-in-phase  in  a  path.  This  implies  that  we  have 
a  measure  for  comparing  different  portions  of  the  path.  We 
shall  suppose  the  time  required  to  traverse  any  portion  of  a 
path  to  be  its  measure  for  the  purpose  of  determining  this 
average. 

With  this  understanding,  let  us  suppose  that  a  certain  en- 
semble is  in  statistical  equilibrium.  In  every  element  of 
extension-in-phase,  therefore,  the  density-in-phase  D  is  equal 
to  its  path-average  27]p.  Let  a  sudden  small  change  be  made 
in  the  external  coordinates.  The  statistical  equilibrium  will  be 
disturbed  and  we  shall  no  longer  have  D  —  ~D\P  everywhere. 
This  is  not  because  D  is  changed,  but  because  ~D\p  is  changed, 
the  paths  being  changed.  It  is  evident  that  if  D  >  I)]p  in 
a  part  of  a  path,  we  shall  have  D  <  ~D\p  in  other  parts  of  the 
same  path. 

Now,  if  we  should  imagine  a  further  change  in  the  external 
coordinates  of  the  same  kind,  we  should  expect  it  to  produce 
an  effect  of  the  same  kind.  But  the  manner  in  which  the 
second  effect  will  be  superposed  on  the  first  will  be  different, 
according  as  it  occurs  immediately  after  the  first  change  or 
after  an  interval  of  time.  If  it  occurs  immediately  after  the 
first  change,  then  in  any  element  of  phase  in  which  the  first 
change  produced  a  positive  value  of  D  -  2J|P  the  second  change 
will  add  a  positive  value  to  the  first  positive  value,  and  where 
D  -  1)\p  was  negative,  the  second  change  will  add  a  negative 
value  to  the  first  negative  value. 

But  if  we  wait  a  sufficient  time  before  making  the  second 
change  in  the  external  coordinates,  so  that  systems  have 
passed  from  elements  of  phase  in  which  D  -  ~D\P  was  origi- 
nally positive  to  elements  in  which  it  was  originally  negative, 
and  vice  versa,  (the  systems  carrying  with  them  the  values 


ON  AN  ENSEMBLE   OF  SYSTEMS.  157 

of  D  -  1J\p ,)  the  positive  values  of  D  -  U\p  caused  by  the 
second  change  will  be  in  part  superposed  on  negative  values 
due  to  the  first  change,  and  vice  versa. 

The  disturbance  of  statistical  equilibrium,  therefore,  pro- 
duced by  a  given  change  in  the  values  of  the  external  co- 
ordinates may  be  very  much  diminished  by  dividing  the 
change  into  two  parts  separated  by  a  sufficient  interval  of 
tune,  and  a  sufficient  interval  of  time  for  this  purpose  is  one 
in  which  the  phases  of  the  individual  systems  are  entirely 
unlike  the  first,  so  that  any  individual  system  is  differently 
affected  by  the  change,  although  the  whole  ensemble  is  af- 
fected in  nearly  the  same  way.  Since  there  is  no  limit  to  the 
diminution  of  the  disturbance  of  equilibrium  by  division  of 
the  change  in  the  external  coordinates,  we  may  suppose  as 
a  general  rule  that  by  diminishing  the  velocity  of  the  changes 
in  the  external  coordinates,  a  given  change  may  be  made  to 
produce  a  very  small  disturbance  of  statistical  equilibrium. 

If  we  write  r[  for  the  value  of  the  average  index  of  probability 
before  the  variation  of  the  external  coordinates,  and  iff'  for  the 
value  after  this  variation,  we  shall  have  in  any  case 


as  the  simple  result  of  the  variation  of  the  external  coordi- 
nates. This  may  be  compared  with  the  thermodynamic  the- 
orem that  the  entropy  of  a  body  cannot  be  diminished  by 
mechanical  (as  distinguished  from  thermal)  action.* 

If  we  have  (approximate)  statistical  equilibrium  between 
the  times  if  and  if'  (corresponding  to  rf  and  ??"),  we  shall  have 
approximately 

which  may  be  compared  with  the  thermodynamic  theorem  that 
the  entropy  of  a  body  is  not  (sensibly)  affected  by  mechanical 
action,  during  which  the  body  is  at  each  instant  (sensibly)  in 
a  state  of  thermodynamic  equilibrium. 

Approximate  statistical  equilibrium  may  usually  be  attained 

*  The  correspondences  to  which  the  reader's  attention  is  called  are  between 
—  t\  and  entropy,  and  between  0  and  temperature. 


158  EFFECT  OF  VARIOUS  PROCESSES 

by  a  sufficiently  slow  variation  of  the  external  coordinates, 
just  as  approximate  thermodynamic  equilibrium  may  usually 
be  attained  by  sufficient  slowness  in  the  mechanical  operations 
to  which  the  body  is  subject. 

We  now  pass  to  the  consideration  of  the  effect  on  an  en- 
semble of  systems  which  is  produced  by  the  action  of  other 
ensembles  with  which  it  is  brought  into  dynamical  connec- 
tion. In  a  previous  chapter  *  we  have  imagined  a  dynamical 
connection  arbitrarily  created  between  the  systems  of  two 
ensembles.  We  shall  now  regard  the  action  between  the 
systems  of  the  two  ensembles  as  a  result  of  the  variation 
of  the  external  coordinates,  which  causes  such  variations 
of  the  internal  coordinates  as  to  bring  the  systems  of  the 
two  ensembles  within  the  range  of  each  other's  action. 

Initially,  we  suppose  that  we  have  two  separate  ensembles 
of  systems,  E±  and  Ez.  The  numbers  of  degrees  of  freedom 
of  the  systems  in  the  two  ensembles  will  be  denoted  by  n^  and 
n2  respectively,  and  the  probability-coefficients  by  e^  and  e"*, 
Now  we  may  regard  any  system  of  the  first  ensemble  com- 
bined with  any  system  of  the  second  as  forming  a  single 
system  of  ^  +  nz  degrees  of  freedom.  Let  us  consider  the 
ensemble  ( J?12)  obtained  by  thus  combining  each  system  of  the 
first  ensemble  with  each  of  the  second. 

At  the  initial  moment,  which  may  be  specified  by  a  single 
accent,  the  probability-coefficient  of  any  phase  of  the  combined 
systems  is  evidently  the  product  of  the  probability-coefficients 
of  the  phases  of  which  it  is  made  up.  This  may  be  expressed 
by  the  equation, 

ew  =  6V  ev,  (455) 

or  n*  =  in'  +  ^  (456) 

which  gives  r^z  =  ij/  +  iya'-  (457) 

The  forces  tending  to  vary  the  internal  coordinates  of  the 
combined  systems,  together  with  those  exerted  by  either 
system  upon  the  bodies  represented  by  the  coordinates  called 

*  See  Chapter  IV,  page  37. 


ON  AN  ENSEMBLE   OF  SYSTEMS.  159 

external,  may  be  derived  from  a  single  force-function,  which, 
taken  negatively,  we  shall  call  the  potential  energy  of  the 
combined  systems  and  denote  by  e12.  But  we  suppose  that 
initially  none  of  the  systems  of  the  two  ensembles  EI  and 
E%  come  within  range  of  each  other's  action,  so  that  the 
potential  energy  of  the  combined  system  falls  into  two  parts 
relating  separately  to  the  systems  which  are  combined.  The 
same  is  obviously  true  of  the  kinetic  energy  of  the  combined 
compound  system,  and  therefore  of  its  total  energy.  This 
may  be  expressed  by  the  equation 

€„'=€/  +  €,',  (458) 

which  gives  e12'  =  i/  +  e2'.  (459) 

Let  us  now  suppose  that  in  the  course  of  tune,  owing  to  the 
motion  of  the  bodies  represented  by  the  coordinates  called 
external,  the  forces  acting  on  the  systems  and  consequently 
their  positions  are  so  altered,  that  the  systems  of  the  ensembles 
El  and  E%  are  brought  within  range  of  each  other's  action, 
and  after  such  mutual  influence  has  lasted  for  a  time,  by  a 
further  change  in  the  external  coordinates,  perhaps  a  return 
to  their  original  values,  the  systems  of  the  two  original  en- 
sembles are  brought  again  out  of  range  of  each  other's  action. 
Finally,  then,  at  a  time  specified  by  double  accents,  we  shall 
have  as  at  first 

€«"  =  e/'  +  ia".  (460) 

But  for  the  indices  of  probability  we  must  write  * 

W  +  W  ^  W'  (461) 

The  considerations  adduced  in  the  last  chapter  show  that  it 
is  safe  to  write 

W  5  W-  (462) 

We  have  therefore 

5i"  +  i"  <  ^  +  i',  (463) 

which  may  be  compared  with  the  thermodynamic  theorem  that 
*  See  Chapter  XI,  Theorem  VII. 


160  EFFECT  OF   VARIOUS  PROCESSES 

the  thermal  contact  of  two  bodies  may  increase  but  cannot 
diminish  the  sum  of  their  entropies. 

Let  us  especially  consider  the  case  in  which  the  two  original 
ensembles  were  both  canonically  distributed  in  phase  with  the 
respective  moduli  ®j  and  ©2.  We  have  then,  by  Theorem  III 
of  Chapter  XI, 

nJ  +  ^  <  r?i"  +  |-  (464) 

^'  +  !'<^"  +  ^  (465) 

Whence  with  (463)  we  have 


_ 

If  we  write  W  for  the  average  work  done  by  the  combined 
systems  on  the  external  bodies,  we  have  by  the  principle  of 
the  conservation  of  energy 

W  =  €„'  -  €M"  =  €/  -  €X"  +  e2'  -  e2".  (468) 

Now  if  TFis  negligible,  we  have 

e/'  _  e/  =  -  (e""  -  €?)  (469) 

and  (467)  shows  that  the  ensemble  which  has  the  greater 
modulus  must  lose  energy.  This  result  may  be  compared  to 
the  thermodynamic  principle,  that  when  two  bodies  of  differ- 
ent temperatures  are  brought  together,  that  which  has  the 
higher  temperature  will  lose  energy. 

Let  us  next  suppose  that  the  ensemble  E%  is  originally 
canonically  distributed  with  the  modulus  @2  ,  but  leave  the 
distribution  of  the  other  arbitrary.  We  have,  to  determine 
the  result  of  a  similar  process, 


ON  AN  ENSEMBLE  OF  SYSTEMS.  161 

Hence  ^"  +  |'=^'  +  C  (470) 

which  may  be  written 

%'-V^^^  (471) 

This  may  be  compared  with  the  thennodynamic  principle  that 
when  a  body  (which  need  not  be  in  thermal  equilibrium)  is 
brought  into  thermal  contact  with  another  of  a  given  tempera- 
ture, the  increase  of  entropy  of  the  first  cannot  be  less  (alge- 
braically) than  the  loss_of  heat  by  the  second  divided  by  its 
temperature.  Where  W  is  negligible,  we  may  write 

V'  +  |^'  +  |  .  (472) 

Now,  by  Theorem  III  of  Chapter  XI,  the  quantity 

!  .  *  +  |  (473) 

has  a  minimum  value  when  the  ensemble  to  which  ^  and  ex 
relate  is  distributed  canonically  with  the  modulus  ®2.  If  the 
ensemble  had  originally  this  distribution,  the  sign  <  in  (472) 
would  be  impossible.  In  fact,  in  this  case,  it  would  be  easy  to 
show  that  the  preceding  formulae  on  which  (472)  is  founded 
would  all  have  the  sign  =  .  But  when  the  two  ensembles  are 
not  both  originally  distributed  canonically  with  the  same 
modulus,  the  formulae  indicate  that  the  quantity  (473)  may 
be  diminished  by  bringing  the  ensemble  to  which  ea  and  yl 
relate  into  connection  with  another  which  is  canonically  dis- 
tributed with  modulus  ®2,  and  therefore,  that  by  repeated 
operations  of  this  kind  the  ensemble  of  which  the  original  dis- 
tribution was  entirely  arbitrary  might  be  brought  approxi- 
mately into  a  state  of  canonical  distribution  with  the  modulus 
<B)2.  We  may  compare  this  with  the  thermodynamic  principle 
that  a  body  of  which  the  original  thermal  state  may  be  entirely 
arbitrary, -may  be  brought  approximately  into  a  state  of  ther- 
mal equilibrium  with  any  given  temperature  by  repeated  con- 
nections with  other  bodies  of  that  temperature. 

11 


162  EFFECT  OF  VARIOUS  PROCESSES 

Let  us  now  suppose  that  we  have  a  certain  number  of 
ensembles,  EQ ,  El ,  E% ,  etc.,  distributed  canonically  with  the 
respective  moduli  ®0 ,  Ox ,  @2 ,  etc.  By  variation  of  the  exter- 
nal coordinates  of  the  ensemble  EQ ,  let  it  be  brought  into 
connection  with  E^ ,  and  then  let  the  connection  be  broken. 
Let  it  then  be  brought  into  connection  with  U2 ,  and  then  let 
that  connection  be  broken.  Let  this  process  be  continued 
with  respect  to  the  remaining  ensembles.  We  do  not  make 
the  assumption,  as  in  some  cases  before,  that  the  work  connected 
with  the  variation  of  the  external  coordinates  is  a  negligible 
quantity.  On  the  contrary,  we  wish  especially  to  consider 
the  case  in  which  it  is  large.  In  the  final  state  of  the  ensem- 
ble EQ  ,  let  us  suppose  that  the  external  coordinates  have  been 
brought  back  to  their  original  values,  and  that  the  average 
energy  (e0)  is  the  same  as  at  first. 

In  our  usual  notations,  using  one  and  two  accents  to  dis- 
tinguish original  and  final  values,  we  get  by  repeated  applica- 
tions of  the  principle  expressed  in  (463) 

V  +  n'  +  V  +  etc.  >  ^0"  +  i"  +  ^2"  +  etc.        (474) 
But  by  Theorem  III  of  Chapter  XI, 


ft"  +        Z  ft1  +      '  (476) 

*"  +  g  >  .7  +  {£  (477) 

etc. 


or,  since  €</  =  €</', 


(479) 

If  we  write  IF  for  the  average  work  done  on  the  bodies  repre- 
sented by  the  external  coordinates,  we  have 


ON  AN  ENSEMBLE   OF  SYSTEMS.  163 

e/  _  e,"  +  ej  -  e2"  +  etc.  =  W.  (480) 

If  EQ,  Ev  and  JE2  are  the  only  ensembles,  we  have 

^<  ©,-_©,  (-,_-,0i  (481) 

It  will  be  observed  that  the  relations  expressed  in  the  last 
three  formulae  between  IF,  ex  —  e/',  e2'  —  e2",  etc.,  and  @1? 
®2,  etc.  are  precisely  those  which  hold  in  a  Carnot's  cycle  for 
the  work  obtained,  the  energy  lost  by  the  several  bodies  which 
serve  as  heaters  or  coolers,  and  their  initial  temperatures. 

It  will  not  escape  the  reader's  notice,  that  while  from  one 
point  of  view  the  operations  which  are  here  described  are  quite 
beyond  our  powers  of  actual  performance,  on  account  of  the 
impossibility  of  handling  the  immense  number  of  systems 
which  are  involved,  yet  from  another  point  of  view  the  opera- 
tions described  are  the  most  simple  and  accurate  means  of 
representing  what  actually  takes  place  in  our  simplest  experi- 
ments in  thermodynamics.  The  states  of  the  bodies  which 
we  handle  are  certainly  not  known  to  us  exactly.  What  we 
know  about  a  body  can  generally  be  described  most  accurately 
and  most  simply  by  saying  that  it  is  one  taken  at  random 
from  a  great  number  (ensemble)  of  bodies  which  are  com- 
pletely described.  If  we  bring  it  into  connection  with  another 
body  concerning  which  we  have  a  similar  limited  knowledge, 
the  state  of  the  two  bodies  is  properly  described  as  that  of  a 
pair  of  bodies  taken  from  a  great  number  (ensemble)  of  pairs 
which  are  formed  by  combining  each  body  of  the  first  en- 
semble with  each  of  the  second. 

Again,  when  we  bring  one  body  into  thermal  contact  with 
another,  for  example,  in  a  Carnot's  cycle,  when  we  bring  a 
mass  of  fluid  into  thermal  contact  with  some  other  body  from 
which  we  wish  it  to  receive  heat,  we  may  do  it  by  moving  the 
vessel  containing  the  fluid.  This  motion  is  mathematically 
expressed  by  the  variation  of  the  coordinates  which  determine 
the  position  of  the  vessel.  We  allow  ourselves  for  the  pur- 
poses of  a  theoretical  discussion  to  suppose  that  the  walls  of 
this  vessel  are  incapable  of  absorbing  heat  from  the  fluid. 


164  EFFECT  OF   VARIOUS  PROCESSES. 

Yet  while  we  exclude  the  kind  of  action  which  we  call  ther- 
mal between  the  fluid  and  the  containing  vessel,  we  allow  the 
kind  which  we  call  work  in  the  narrower  sense,  which  takes 
place  when  the  volume  of  the  fluid  is  changed  by  the  motion 
of  a  piston.  This  agrees  with  what  we  have  supposed  in 
regard  to  the  external  coordinates,  which  we  may  vary  in 
any  arbitrary  manner,  and  are  in  this  entirely  unlike  the  co- 
ordinates of  the  second  ensemble  with  which  we  bring  the 
first  into  connection. 

When  heat  passes  in  any  thermodynamic  experiment  between 
the  fluid  principally  considered  and  some  other  body,  it  is 
actually  absorbed  and  given  out  by  the  walls  of  the  vessel, 
which  will  retain  a  varying  quantity.  This  is,  however,  a 
disturbing  circumstance,  which  we  suppose  in  some  way  made 
negligible,  and  actually  neglect  in  a  theoretical  discussion. 
In  our  case,  we  suppose  the  walls  incapable  of  absorbing  en- 
ergy, except  through  the  motion  of  the  external  coordinates, 
but  that  they  allow  the  systems  which  they  contain  to  act 
directly  on  one  another.  Properties  of  this  kind  are  mathe- 
matically expressed  by  supposing  that  in  the  vicinity  of  a 
certain  surface,  the  position  of  which  is  determined  by  certain 
(external)  coordinates,  particles  belonging  to  the  system  in 
question  experience  a  repulsion  from  the  surface  increasing  so 
rapidly  with  nearness  to  the  surface  that  an  infinite  expendi- 
ture of  energy  would  be  required  to  carry  them  through  it. 
It  is  evident  that  two  systems  might  be  separated  by  a  surface 
or  surfaces  exerting  the  proper  forces,  and  yet  approach  each 
other  closely  enough  to  exert  mechanical  action  on  each  other. 


CHAPTER  XIV. 

DISCUSSION  OF  THERMODYNAMIC   ANALOGIES. 

IF  we  wish  to  find  in  rational  mechanics  an  a  priori  founda- 
tion for  the  principles  of  thermodynamics,  we  must  seek 
mechanical  definitions  of  temperature  and  entropy.  The 
quantities  thus  defined  must  satisfy  (under  conditions  and 
with  limitations  which  again  must  be  specified  in  the  language 
of  mechanics)  the  differential  equation 

de  =  Td-q  —  Al  dal  —  A2  daz  —  etc.,  (482) 

where  e,  T,  and  TJ  denote  the  energy,  temperature,  and  entropy 
of  the  system  considered,  and  A^dav  etc.,  the  mechanical  work 
(in  the  narrower  sense  in  which  the  term  is  used  in  thermo- 
dynamics, i.  e.,  with  exclusion  of  thermal  action)  done  upon 
external  bodies. 

This  implies  that  we  are  able  to  distinguish  in  mechanical 
terms  the  thermal  action  of  one  system  on  another  from  that 
which  we  call  mechanical  in  the  narrower  sense,  if  not  indeed 
in  every  case  hi  which  the  two  may  be  combined,  at  least  so  as 
to  specify  cases  of  thermal  action  and  cases  of  mechanical 
action. 

Such  a  differential  equation  moreover  implies  a  finite  equa- 
tion between  e,  ?/,  and  av  a2,  etc.,  which  may  be  regarded 
as  fundamental  in  regard  to  those  properties  of  the  system 
which  we  call  thermodynamic,  or  which  may  be  called  so  from 
analogy.  This  fundamental  thermodynamic  equation  is  de- 
termined by  the  fundamental  mechanical  equation  which 
expresses  the  energy  of  the  system  as  function  of  its  mo- 
menta and  coordinates  with  those  external  coordinates  (av  «2, 
etc.)  which  appear  in  the  differential  expression  of  the  work 
done  on  external  bodies.  We  have  to  show  the  mathematical 
operations  by  which  the  fundamental  thermodynamic  equation, 


166  THERMODYNAMIC  ANALOGIES. 

which  in  general  is  an  equation  of  few  variables,  is  derived 
from  the  fundamental  mechanical  equation,  which  in  the  case 
of  the  bodies  of  nature  is  one  of  an  enormous  number  of 
variables. 

We  have  also  to  enunciate  in  mechanical  terms,  and  to 
prove,  what  we  call  the  tendency  of  heat  to  pass  from  a  sys- 
tem of  higher  temperature  to  one  of  lower,  and  to  show  that 
this  tendency  vanishes  with  respect  to  systems  of  the  same 
temperature. 

At  least,  we  have  to  show  by  a  priori  reasoning  that  for 
such  systems  as  the  material  bodies  which  nature  presents  to 
us,  these  relations  hold  with  such  approximation  that  they 
are  sensibly  true  for  human  faculties  of  observation.  This 
indeed  is  all  that  is  really  necessary  to  establish  the  science  of 
thermodynamics  on  an  a  priori  basis.  Yet  we  will  naturally 
desire  to  find  the  exact  expression  of  those  principles  of  which 
the  laws  of  thermodynamics  are  the  approximate  expression. 
A  very  little  study  of  the  statistical  properties  of  conservative 
systems  of  a  finite  number  of  degrees  of  freedom  is  sufficient 
to  make  it  appear,  more  or  less  distinctly,  that  the  general 
laws  of  thermodynamics  are  the  limit  toward  which  the  exact 
laws  of  such  systems  approximate,  when  their  number  of 
degrees  of  freedom  is  indefinitely  increased.  And  the  problem 
of  finding  the  exact  relations,  as  distinguished  from  the  ap- 
proximate, for  systems  of  a  great  number  of  degrees  of  free- 
dom, is  practically  the  same  as  that  of  finding  the  relations 
which  hold  for  any  number  of  degrees  of  freedom,  as  distin- 
guished from  those  which  have  been  established  on  an  em- 
pirical basis  for  systems  of  a  great  number  of  degrees  of 
freedom. 

The  enunciation  and  proof  of  these  exact  laws,  for  systems 
of  any  finite  number  of  degrees  of  freedom,  has  been  a  princi- 
pal object  of  the  preceding  discussion.  But  it  should  be  dis- 
tinctly stated  that,  if  the  results  obtained  when  the  numbers 
of  degrees  of  freedom  are  enormous  coincide  sensibly  with 
the  general  laws  of  thermodynamics,  however  interesting  and 
significant  this  coincidence  may  be,  we  are  still  far  from 


THERMODYNAMIC  ANALOGIES.  167 

having  explained  the  phenomena  of  nature  with  respect  to 
these  laws.  For,  as  compared  with  the  case  of  nature,  the 
systems  which  we  have  considered  are  of  an  ideal  simplicity. 
Although  our  only  assumption  is  that  we  are  considering 
conservative  systems  of  a  finite  number  of  degrees  of  freedom, 
it  would  seem  that  this  is  assuming  far  too  much,  so  far  as  the 
bodies  of  nature  are  concerned.  The  phenomena  of  radiant 
heat,  which  certainly  should  not  be  neglected  in  any  complete 
system  of  thermodynamics,  and  the  electrical  phenomena 
associated  with  the  combination  of  atoms,  seem  to  show  that 
the  hypothesis  of  systems  of  a  finite  number  of  degrees  of 
freedom  is  inadequate  for  the  explanation  of  the  properties  of 
bodies. 

Nor  do  the  results  of  such  assumptions  in  every  detail 
appear  to  agree  with  experience.  We  should  expect,  for 
example,  that  a  diatomic  gas,  so  far  as  it  could  be  treated 
independently  of  the  phenomena  of  radiation,  or  of  any  sort  of 
electrical  manifestations,  would  have  six  degrees  of  freedom 
for  each  molecule.  But  the  behavior  of  such  a  gas  seems  to 
indicate  not  more  than  five. 

But  although  these  difficulties,  long  recognized  by  physi- 
cists,* seem  to  prevent,  in  the  present  state  of  science,  any 
satisfactory  explanation  of  the  phenomena  of  thermodynamics 
as  presented  to  us  in  nature,  the  ideal  case  of  systems  of  a 
finite  number  of  degrees  of  freedom  remains  as  a  subject 
which  is  certainly  not  devoid  of  a  theoretical  interest,  and 
which  may  serve  to  point  the  way  to  the  solution  of  the  far 
more  difficult  problems  presented  to  us  by  nature.  And  if 
the  study  of  the  statistical  properties  of  such  systems  gives 
us  an  exact  expression  of  laws  which  in  the  limiting  case  take 
the  form  of  the  received  laws  of  thermodynamics,  its  interest 
is  so  much  the  greater. 

Now  we  have  defined  what  we  have  called  the  modulus  (O) 
of  an  ensemble  of  systems  canonically  distributed  in  phase, 
and  wha't  we  have  called  the  index  of  probability  (77)  of  any 
phase  in  such  an  ensemble.  It  has  been  shown  that  between 

*  See  Boltzmann,  Sitzb.  der  Wiener  Akad.,  Bd.  LXIIL,  S.  418,  (1871). 


168  THERMODYNAMIC  ANALOGIES. 

the  modulus  (@),  the  external  coordinates  (al9  etc.),  and  the 
average  values  in  the  ensemble  of  the  energy  (e),  the  index 
of  probability  (?;),  and  the  external  forces  (A19  etc.)  exerted 
by  the  systems,  the  following  differential  equation  will  hold  : 


cfe  =  —  ©  dj  —  Al  da-L  —  JT2  da2  —  etc.  (483) 

This  equation,  if  we  neglect  the  sign  of  averages,  is  identical 
in  form  with  the  thermodynamic  equation  (482),  the  modulus 
(®)  corresponding  to  temperature,  and  the  index  of  probabil- 
ity of  phase  with  its  sign  reversed  corresponding  to  entropy.* 

We  have  also  shown  that  the  average  square  of  the  anoma- 
lies of  e,  that  is,  of  the  deviations  of  the  individual  values  from 
the  average,  is  in  general  of  the  same  order  of  magnitude  as 
the  reciprocal  of  the  number  of  degrees  of  freedom,  and  there- 
fore to  human  observation  the  individual  values  are  indistin- 
guishable from  the  average  values  when  the  number  of  degrees 
of  freedom  is  very  great.  f  In  this  case  also  the  anomalies  of  q 
are  practically  insensible.  The  same  is  true  of  the  anomalies  of 
the  external  forces  (A^  ,  etc.),  so  far  as  these  are  the  result  of 
the  anomalies  of  energy,  so  that  when  these  forces  are  sensibly 
determined  by  the  energy  and  the  external  coordinates,  and 
the  number  of  degrees  of  freedom  is  very  great,  the  anomalies 
of  these  forces  are  insensible. 

The  mathematical  operations  by  which  the  finite  equation 
between  e,  77,  and  ax  ,  etc.,  is  deduced  from  that  which  gives 
the  energy  (e)  of  a  system  in  terms  of  the  momenta  (j)l  .  .  .  .pn) 
and  coordinates  both  internal  (^  .  .  .  <?„)  and  external  (ax  ,  etc.), 
are  indicated  by  the  equation 

$  all  € 

\  e~&  =f.  .  .§e~®dq,  .  .  .  dqndp,  .  .  .  dpn,         (484) 

phases 

where  ^  =  ®rj  +  e. 

We  have  also  shown  that  when  systems  of  different  ensem- 
bles are  brought  into  conditions  analogous  to  thermal  contact, 
the  average  result  is  a  passage  of  energy  from  the  ensemble 

*  See  Chapter  IV,  pages  44,  45.  t  See  Chapter  VII,  pages  73-75. 


THERMODYNAMIC  ANALOGIES.  169 

of  the  greater  modulus  to  that  of  the  less,  *  or  in  case  of  equal 
moduli,  that  we  have  a  condition  of  statistical  equilibrium  in 
regard  to  the  distribution  of  energy,  f 

Propositions  have  also  been  demonstrated  analogous  to 
those  in  thermodynamics  relating  to  a  Carnot's  cycle,:]:  or  to 
the  tendency  of  entropy  to  increase,  §  especially  when  bodies 
of  different  temperature  are  brought  into  contact.  || 

We  have  thus  precisely  defined  quantities,  and  rigorously 
demonstrated  propositions,  which  hold  for  any  number  of 
degrees  of  freedom,  and  which,  when  the  number  of  degrees 
of  freedom  (n)  is  enormously  great,  would  appear  to  human 
faculties  as  the  quantities  and  propositions  of  empirical  ther- 
modynamics. 

It  is  evident,  however,  that  there  may  be  more  than  one 
quantity  denned  for  finite  values  of  n,  which  approach  the 
same  limit,  when  n  is  increased  indefinitely,  and  more  than  one 
proposition  relating  to  finite  values  of  n,  which  approach  the 
same  limiting  form  for  n  =  oo.  There  may  be  therefore, 
and  there  are,  other  quantities  which  may  be  thought  to  have 
some  claim  to  be  regarded  as  temperature  and  entropy  with 
respect  to  systems  of  a  finite  number  of  degrees  of  freedom. 

The  definitions  and  propositions  which  we  have  been  con- 
sidering relate  essentially  to  what  we  have  called  a  canonical 
ensemble  of  systems.  This  may  appear  a  less  natural  and 
simple  conception  than  what  we  have  called  a  microcanonical 
ensemble  of  systems,  in  which  all  have  thex-sa"me^energyvand 
which  in  many  cases  represents  simply  tltte  time-ensemble,  or 
ensemble  of  phases  through  which  a  single  system  passes  in 
the  course  of  time. 

It  may  therefore  seem  desirable  to  find  definitions  and 
propositions  relating  to  these  microcanonical  ensembles,  which 
shall  correspond  to  what  in  thermodynamics  are  based  on 
experience.  Now  the  differential  equation 

de  =  e~*  Vd  log  F-  ZTle  «fai  -  3^]6  daz  -  etc.,       (485) 


*  See  Chapter  XIII,  page  160.  t  See  Chapter  IV,  pages  35-37. 

J  See  Chapter  XIII,  pages  162,  163.       §  See  Chapter  XII,  pages  143-151. 
||  See  Chapter  XIII,  page  159. 


170  THERMODYNAMIC  ANALOGIES. 

which  has  been  demonstrated  in  Chapter  X,  and  which  relates  to 
a  microcanonical  ensemble,  A^  denoting  the  average  value  of 
A1  in  such  an  ensemble,  corresponds  precisely  to  the  thermody- 
namic  equation,  except  for  the  sign  of  average  applied  to  the 
external  forces.  But  as  these  forces  are  not  entirely  deter- 
mined by  the  energy  with  the  external  coordinates,  the  use  of 
average  values  is  entirely  germane  to  the  subject,  and  affords 
the  readiest  means  of  getting  perfectly  determined  quantities. 
These  averages,  which  are  taken  for  a  microcanonical  ensemble, 
may  seem  from  some  points  of  view  a  more  simple  and  natural 
conception  than  those  which  relate  to  a  canonical  ensemble. 
Moreover,  the  energy,  and  the  quantity  corresponding  to  en- 
tropy, are  free  from  the  sign  of  average  in  this  equation. 

The  quantity  in  the  equation  which  corresponds  to  entropy 
is  log  FJ  the  quantity  V  being  defined  as  the  extension-in- 
phase  within  which  the  energy  is  less  than  a  certain  limiting 
value  (e).  This  is  certainly  a  more  simple  conception  than  the 
average  value  in  a  canonical  ensemble  of  the  index  of  probabil- 
ity of  phase.  Log  V  has  the  property  that  when  it  is  constant 

de  =  -  21].  dat  -  A^\f  daz  +  etc.,  (486) 

which  closely  corresponds  to  the  thermodynamic  property  of 
entropy,  that  when  it  is  constant 

de  =  —  Aj_  da^  —  A2  daz  +  etc.  (487) 

The  quantity  in  the  equation  which  corresponds  to  tem- 
perature is  e~*  F",  or  dejd  log  V.  In  a  canonical  ensemble,  the 
average  value  of  this  quantity  is  equal  to  the  modulus,  as  has 
been  shown  by  different  methods  in  Chapters  IX  and  X. 

In  Chapter  X  it  has  also  been  shown  that  if  the  systems 
of  a  microcanonical  ensemble  consist  of  parts  with  separate 
energies,  the  average  value  of  e~*  Vi or  any  part  is  equal  to  its 
average  value  for  any  other  part,  and  to  the  uniform  value 
of  the  same  expression  for  the  whole  ensemble.  This  corre- 
sponds to  the  theorem  in  the  theory  of  heat  that  in  case  of 
thermal  equilibrium  the  temperatures  of  the  parts  of  a  body 
are  equal  to  one  another  and  to  that  of  the  whole  body. 


THERMODYNAMIC  ANALOGIES.  171 

Since  the  energies  of  the  parts  of  a  body  cannot  be  supposed 
to  remain  absolutely  constant,  even  where  this  is  the  case 
with  respect  to  the  whole  body,  it  is  evident  that  if  we  regard 
the  temperature  as  a  function  of  the  energy,  the  taking  of 
average  or  of  probable  values,  or  some  other  statistical  process, 
must  be  used  with  reference  to  the  parts,  in  order  to  get  a 
perfectly  definite  value  corresponding  to  the  notion  of  tem- 
perature. 

It  is  worthy  of  notice  in  this  connection  that  the  average 
value  of  the  kinetic  energy,  either  in  a  microcanonical  en- 
semble, or  in  a  canonical,  divided  by  one  half  the  number  of 
degrees  of  freedom,  is  equal  to  e~*  "FJ  or  to  its  average  value, 
and  that  this  is  true  not  only  of  the  whole  system  which  is 
distributed  either  microcanonically  or  canonically,  but  also 
of  any  part,  although  the  corresponding  theorem  relating  to 
temperature  hardly  belongs  to  empirical  thermodynamics,  since 
neither  the  (inner)  kinetic  energy  of  a  body,  nor  its  number 
of  degrees  of  freedom  is  immediately  cognizable  to  our  facul- 
ties, and  we  meet  the  gravest  difficulties  when  we  endeavor 
to  apply  the  theorem  to  the  theory  of  gases,  except  in  the 
simplest  case,  that  of  the  gases  known  as  monatomic. 

But  the  correspondence  between  &~*  V  or  dejd  log  V  and 
temperature  is  imperfect.  If  two  isolated  systems  have  such 
energies  that 

de-L  de2 

d  log  FI  ~~  d  log  F2  ' 

and  the  two  systems  are  regarded  as  combined  to  form  a  third 
system  with  energy 

€12  =  ex  +  e2> 
we  shall  not  have  in  general 

deiz  del  dez 


dlog  F12  ~~  dlog  Fi  ~  dlog  F2' 

as  analogy  with  temperature  would  require.     In  fact,  we  have 
seen  that 


d  log  F12      d  log  FitM  ~~  d  log  Fj 


172  THERMODYNAMIC  ANALOGIES. 

where  the  second  and  third  members  of  the  equation  denote 
average  values  in  an  ensemble  in  which  the  compound  system 
is  microcanonically  distributed  in  phase.  Let  us  suppose  the 
two  original  systems  to  be  identical  in  nature.  Then 


The  equation  in  question  would  require  that 


i.  e.,  that  we  get  the  same  result,  whether  we  take  the  value 
of  del/dlog  V}  determined  for  the  average  value  of  e1  in  the 
ensemble,  or  take  the  average  value  of  de^dlog  F"r  This 
will  be  the  case  where  de^dlog  V^  is  a  linear  function  of  er 
Evidently  this  does  not  constitute  the  most  general  case. 
Therefore  the  equation  in  question  cannot  be  true  in  general. 
It  is  true,  however,  in  some  very  important  particular  cases,  as 
when  the  energy  is  a  quadratic  function  of  the  p's  and  ^'s,  or 
of  the  p's  alone.*  When  the  equation  holds,  the  case  is  anal- 
ogous to  that  of  bodies  in  thermodynamics  for  which  the 
specific  heat  for  constant  volume  is  constant. 

Another  quantity  which  is  closely  related  to  temperature  is 
dcfr/de.  It  has  been  shown  in  Chapter  IX  that  in  a  canonical 
ensemble,  if  n  >  2,  the  average  value  of  d(f>fde  is  I/®,  and 
that  the  most  common  value  of  the  energy  in  the  ensemble  is 
that  for  which  d$/de  =  I/®.  The  first  of  these  properties 
may  be  compared  with  that  of  de/dlog  V,  which  has  been 
seen  to  have  the  average  value  ®  in  a  canonical  ensemble, 
without  restriction  in  regard  to  the  number  of  degrees  of 
freedom. 

With  respect  to  microcanonical  ensembles  also,  dfyjde  has 
a  property  similar  to  what  has  been  mentioned  with  respect  to 
de/d  log  V.  That  is,  if  a  system  microcanonically  distributed 
in  phase  consists  of  two  parts  with  separate  energies,  and  each 

*  This  last  case  is  important  on  account  of  its  relation  to  the  theory  of 
gases,  although  it  must  in  strictness  be  regarded  as  a  limit  of  possible  cases, 
rather  than  as  a  case  which  is  itself  possible. 


THERMODYNAMIC  ANALOGIES.  173 

with  more  than  two  degrees  of  freedom,  the  average  values  in 
the  ensemble  of  d(f)/de  for  the  two  parts  are  equal  to  one 
another  and  to  the  value  of  same  expression  for  the  whole. 
In  our  usual  notations 


"^12 

de2  L2  ~"  delz 


if  TIX  >  2,  and  n2  >  2. 

This  analogy  with  temperature  has  the  same  incompleteness 
which  was  noticed  with  respect  to  de/dlog  V,  viz.,  if  two  sys- 
tems have  such  energies  (ej  and  e2)  that 


and  they  are  combined  to  form  a  third  system  with  energy 

*ia  =  €1  +  €2, 

we  shall  not  have  in  general 

c?012  _  dfa  __  d<f>z 
d€i2       deL       dez  ' 

Thus,  if  the  energy  is  a  quadratic  function  of  the  p's  and  <?'s, 
we  have  * 


e12  €la  el  +  e2 

where  nt,  w2,  w12,  are  the  numbers  of  degrees  of  freedom  of  the 
separate  and  combined  systems.  But 

dfa      d<f>2      HI  +  n%  —  2 
del  ~  de2  "        e1  +  ez 

If  the  energy  is  a  quadratic  function  of  the  p's  alone,  the  case 
would  be  the  same  except  that  we  should  have  J  n^  ,  J  w2  ,  J  w12  , 
instead  of  wx  ,  w2  ,  w12.  In  these  particular  cases,  the  analogy 

*  See  foot-note  on  page  93.  We  have  here  made  the  least  value  of  the 
energy  consistent  with  the  values  of  the  external  coordinates  zero  instead 
of  ea,  as  is  evidently  allowable  when  the  external  coordinates  are  supposed 
invariable. 


174  THERMODYNAMIC  ANALOGIES. 

between  de/d  log  V  and  temperature  would  be  complete,  as  has 
already  been  remarked.     We  should  have 

del          e^  c?62       _  e2 

n'9  dlo    V~' 


_= 
MM       rflog  F!       dlogV2' 

when  the  energy  is  a  quadratic  function  of  the  p's  and  #'s,  and 
similar  equations  with  £  %  ,  J  ra2  ,  -|-  w12  ,  instead  of  ^  ,  w2  ,  w12  , 
when  the  energy  is  a  quadratic  function  of  the  £>'s  alone. 

More  characteristic  of  dcf>/de  are  its  properties  relating  to 
most  probable  values  of  energy.  If  a  system  having  two  parts 
with  separate  energies  and  each  with  more  than  two  degrees 
of  freedom  is  microcanonically  distributed  in  phase,  the  most 
probable  division  of  energy  between  the  parts,  in  a  system 
taken  at  random  from  the  ensemble,  satisfies  the  equation 

^  =  ^,  (488) 

del       de2 

which  corresponds  to  the  thermodynamic  theorem  that  the 
distribution  of  energy  between  the  parts  of  a  system,  in  case  of 
thermal  equilibrium,  is  such  that  the  temperatures  of  the  parts 
are  equal. 

To  prove  the  theorem,  we  observe  that  the  fractional  part 
of  the  whole  number  of  systems  which  have  the  energy  of  one 
part  (ej)  between  the  limits  e/  and  e/  is  expressed  by 

r*f.  ****>, 

T  i 

where  the  variables  are  connected  by  the  equation 
€j  -|-  €2  =  constant  =  ei2  . 

The  greatest  value  of  this  expression,  for  a  constant  infinitesi- 
mal value  of  the  difference  ex"  —  e/,  determines  a  value  of  e1  , 
which  we  may  call  its  most  probable  value.  This  depends  on 
the  greatest  possible  value  of  fa  +  fa.  Now  if  n^  >  2,  and 
w2  >  2,  we  shall  have  fa  =  —  oo  for  the  least  possible  value  of 


THERMODYNAMIC  ANALOGIES.  175 

6j  ,  and  <f>2  =  —  QO  for  the  least  possible  value  of  e2.  Between 
these  limits  (/>x  and  <£2  will  be  finite  and  continuous.  Hence 
$!  +  <£2  will  have  a  maximum  satisfying  the  equation  (488). 

But  if  n^  <  2,  or  w2  <  2,  d(f)1/d€l  or  d$2/de2  may  be  nega- 
tive, or  zero,  for  all  values  of  e1  or  e2,  and  can  hardly  be 
regarded  as  having  properties  analogous  to  temperature. 

It  is  also  worthy  of  notice  that  if  a  system  which  is  micro- 
canonically  distributed  in  phase  has  three  parts  with  separate 
energies,  and  each  with  more  than  two  degrees  of  freedom,  the 
most  probable  division  of  energy  between  these  parts  satisfies 
the  equation 


That  is,  this  equation  gives  the  most  probable  set  of  values 
of  ej,  62,  and  e3.  But  it  does  not  give  the  most  probable 
value  of  el  ,  or  of  e2  ,  or  of  e3.  Thus,  if  the  energies  are  quad- 
ratic functions  of  the  p9s  and  <?'s,  the  most  probable  division 
of  energy  is  given  by  the  equation 

HI  —  1  _  n<2,  —  1  _  nz  —  1 

«i  «i  €« 

But  the  most  probable  value  of  ei  is  given  by 


while  the  preceding  equations  give 
KI  —  1      ^2  + 

«1 

These  distinctions  vanish  for  very  great  values  of  n^  ,  n2  ,  w3. 
For  small  values  of  these  numbers,  they  are  important.  Such 
facts  seem  to  indicate  that  the  consideration  of  the  most 
probable  division  of  energy  among  the  parts  of  a  system  does 
not  afford  a  convenient  foundation  for  the  study  of  thermody- 
namic  analogies  in  the  case  of  systems  of  a  small  number  of 
degrees  of  'freedom.  The  fact  that  a  certain  division  of  energy 
is  the  most  probable  has  really  no  especial  physical  importance, 
except  when  the  ensemble  of  possible  divisions  are  grouped  so 


176  THERMODYNAMIC  ANALOGIES. 

closely  together  that  the  most  probable  division  may  fairly 
represent  the  whole.  This  is  in  general  the  case,  to  a  very 
close  approximation,  when  n  is  enormously  great  ;  it  entirely 
fails  when  n  is  small. 

If  we  regard  dcfr/de  as  corresponding  to  the  reciprocal  of 
temperature,  or,  in  other  words,  de/d(f>  as  corresponding  to 
temperature,  <£  will  correspond  to  entropy.  It  has  been  denned 
as  log  (d  V/de).  In  the  considerations  on  which  its  definition 
is  founded,  it  is  therefore  very  similar  to  log  F".  We  have 
seen  that  d(j>/dlogV  approaches  the  value  unity  when  n  is 
very  great.*  . 

To  form  a  differential  equation  on  the  model  of  the  thermo- 
dynamic  equation  (482),  in  which  de/dcf)  shall  take  the  place 
of  temperature,  and  <£  of  entropy,  we  may  write 

da*  +  etc->  (489> 


or  <Z*=         de  +        da1  +        da2  +  ete.  (490) 

de  da-L  da2 

With  respect  to  the  differential  coefficients  in  the  last  equa- 
tion, which  corresponds  exactly  to  (482)  solved  with  respect 
to  drj9  we  have  seen  that  their  average  values  in  a  canonical 
ensemble  are  equal  to  I/®,  and  the  averages  of  Al/®,  A2/®, 
etc.f  We  have  also  seen  that  de/dcfr  (or  d(f>/de)  has  relations 
to  the  most  probable  values  of  energy  in  parts  of  a  microca- 
nonical  ensemble.  That  (del  da^)^,  etc.,  have  properties 
somewhat  analogous,  may  be  shown  as  follows. 

In  a  physical  experiment,  we  measure  a  force  by  balancing  it 
against  another.  If  we  should  ask  what  force  applied  to  in- 
crease or  diminish  &x  would  balance  the  action  of  the  systems, 
it  would  be  one  which  varies  with  the  different  systems.  But 
we  may  ask  what  single  force  will  make  a  given  value  of  a^ 
the  most  probable,  and  we  shall  find  that  under  certain  condi- 
tions (de/da^Q,  a  represents  that  force. 

*  See  Chapter  X,  pages  120,  121. 

t  See  Chapter  IX,  equations  (321),  (327). 


THERMODYNAMIC  ANALOGIES.  177 

To  make  the  problem  definite,  let  us  consider  a  system  con- 
sisting of  the  original  system  together  with  another  having 
the  coordinates  a^  ,  a2  ,  etc.,  and  forces  AJ,  A<£,  etc.,  tending 
to  increase  those  coordinates.  These  are  in  addition  to  the 
forces  Av  Av  etc.,  exerted  by  the  original  system,  and  are  de- 
rived from  a  force-function  (—  eg')  by  the  equations 

^;_       &J  A,-   _^L         etc 

Al  '     ~d^>  da2' 

For  the  energy  of  the  whole  system  we  may  write 
E  =  e  +  ej  +  •Jm1a'12  +  im2a22  +  etc., 

and  for  the  extension-in-phase  of  the  whole  system  within  any 
limits 

I  ...  I  dpi  .  .  .  dqn  da,i  mi  da±  daz  mz  da2  .  .  . 

or  I  .  .  .  I  e$  de  da-^  m1  da^  daz  m2  da2  .  .  .  , 

or  again  I  .  .  .  /  e^  d&  dat  mx  dai  da2  m2  da2  .  .  .  , 

since  de  =  c?E,  when  ax,  ax,  a2,  «2,  etc.,  are  constant.  If  the 
limits  are  expressed  by  E  and  E  +  c?E,  a^  and  a-^  +  da^  ,  a1  and 
«j  +  da-^  ,  etc.,  the  integral  reduces  to 


The  values  of  ^  ,  ax  ,  «2  ,  <z2  ,  etc.,  which  make  this  expression 
a  maximum  for  constant  values  of  the  energy  of  the  whole 
system  and  of  the  differentials  dE,  da19  dal9  etc.,  are  what  may 
be  called  the  most  probable  values  of  ax  ,  a^  ,  etc.,  in  an  ensem- 
ble in  which  the  whole  system  is  distributed  microcanonieally. 
To  determine  these  values  we  have 

de*  =  0, 
when  d(e  +  eq'  +  i  m  of  +  i  m2  «22  +  etc.)  =  0. 

That  is,  d$  —  0, 

12 


178  THERMODYNAMIC  ANALOGIES. 


when 

(—  - 
«<£ 


etc.  +  m-^  a\  dat  +  etc.  =  0. 


This  requires  %  =  0,    az  =  0,    etc., 

and  (-vM      =^i,     fir]      =  A',    etc. 

\««i/^,a  \da*J^a 

This   shows   that  for  any  given  values   of   E,   aj,  a2,    etc. 

(  -7—  )  ,  (  -:—  )  ,  etc.,  represent  the  forces  (in  the  oren- 
\aai/*,a  \dazj^a 

eralized  sense)  which  the  external  bodies  would  have  to  exert 
to  make  these  values  of  a^  ,  «2  ,  etc.,  the  most  probable  under 
the  conditions  specified.  When  the  differences  of  the  external 
forces  which  are  exerted  by  the  different  systems  are  negli- 
gible, —  (d€/da^)$tm  etc.,  represent  these  forces. 

It  is  certainly  in  the  quantities  relating  to  a  canonical 
ensemble,  e,  ®,  ??,  JL1?  etc.,  ax,  etc.,  that  we  find  the  most 
complete  correspondence  with  the  quantities  of  the  thermody- 
namic  equation  (482).  Yet  the  conception  itself  of  the  canon- 
ical ensemble  may  seem  to  some  artificial,  and  hardly  germane 
to  a  natural  exposition  of  the  subject;  and  the  quantities 
de  .  Tr  -,—  .  de  de 

°S  v>  a.  ete"  «i.  etc"  ore'       - 


etc.,  flj,  etc.,  which  are  closely  related  to  ensembles  of  constant 
energy,  and  to  average  and  most  probable  values  in  such 
ensembles,  and  most  of  which  are  defined  without  reference 
to  any  ensemble,  may  appear  the  most  natural  analogues  of 
the  thermodynamic  quantities. 

In  regard  to  the  naturalness  of  seeking  analogies  with  the 
thermodynamic  behavior  of  bodies  in  canonical  or  microca- 
nonical  ensembles  of  systems,  much  will  depend  upon  how  we 
approach  the  subject,  especially  upon  the  question  whether  we 
regard  energy  or  temperature  as  an  independent  variable. 

It  is  very  natural  to  take  energy  for  an  independent  variable 
rather  than  temperature,  because  ordinary  mechanics  furnishes 
us  with  a  perfectly  defined  conception  of  energy,  whereas  the 
idea  of  something  relating  to  a  mechanical  system  and  corre- 


THERMODYNAMIC  ANALOGIES.  179 

spending  to  temperature  is  a  notion  but  vaguely  denned.  Now 
if  the  state  of  a  system  is  given  by  its  energy  and  the  external 
coordinates,  it  is  incompletely  denned,  although  its  partial  defi- 
nition is  perfectly  clear  as  far  as  it  goes.  The  ensemble  of 
phases  microcanonically  distributed,  with  the  given  values  of 
the  energy  and  the  external  coordinates,  will  represent  the  im- 
perfectly defined  system  better  than  any  other  ensemble  or 
single  phase.  When  we  approach  the  subject  from  this  side, 
our  theorems  will  naturally  relate  to  average  values,  or  most 
probable  values,  in  such  ensembles. 

In  this  case,  the  choice  between  the  variables  of  (485)  or  of 
(489)  will  be  determined  partly  by  the  relative  importance 
which  is  attached  to  average  and  probable  values.  It  would 
seem  that  in  general  average  values  are  the  most  important,  and 
that  they  lend  themselves  better  to  analytical  transformations. 
This  consideration  would  give  the  preference  to  the  system  of 
variables  in  which  log  V  is  the  analogue  of  entropy.  Moreover, 
if  we  make  <f>  the  analogue  of  entropy,  we  are  embarrassed  by 
the  necessity  of  making  numerous  exceptions  for  systems  of 
one  or  two  degrees  of  freedom. 

On  the  other  hand,  the  definition  of  <£  may  be  regarded  as  a 
little  more  simple  than  that  of  log  F",  and  if  our  choice  is  deter- 
mined by  the  simplicity  of  the  definitions  of  the  analogues  of 
entropy  and  temperature,  it  would  seem  that  the  <£  system 
should  have  the  preference.  In  our  definition  of  these  quanti- 
ties, V  was  defined  first,  and  e^  derived  from  V  by  differen- 
tiation. This  gives  the  relation  of  the  quantities  in  the  most 
simple  analytical  form.  Yet  so  far  as  the  notions  are  con- 
cerned, it  is  perhaps  more  natural  to  regard  Fas  derived  from 
C*  by  integration.  At  all  events,  e*  may  be  defined  inde- 
pendently of  F",  and  its  definition  niay  be  regarded  as  more 
simple  as  not  requiring  the  determination  of  the  zero  from 
which  V  is  measured,  which  sometimes  involves  questions 
of  a  delicate  nature.  In  fact,  the  quantity  e*  may  exist, 
when  the  definition  of  V  becomes  illusory  for  practical  pur- 
poses, as  the  integral  by  which  it  is  determined  becomes  infinite. 

The  case  is  entirely  different,  when  we  regard  the  tempera- 


180  THERMODYNAMIC  ANALOGIES. 

ture  as  an  independent  variable,  and  we  have  to  consider  a 
system  which  is  described  as  having  a  certain  temperature  and 
certain  values  for  the  external  coordinates.  Here  also  the 
state  of  the  system  is  not  completely  denned,  and  will  be 
better  represented  by  an  ensemble  of  phases  than  by  any  single 
phase.  What  is  the  nature  of  such  an  ensemble  as  will  best 
represent  the  imperfectly  defined  state  ? 

When  we  wish  to  give  a  body  a  certain  temperature,  we 
place  it  in  a  bath  of  the  proper  temperature,  and  when  we 
regard  what  we  call  thermal  equilibrium  as  established,  we  say 
that  the  body  has  the  same  temperature  as  the  bath.  Per- 
haps we  place  a  second  body  of  standard  character,  which  we 
call  a  thermometer,  in  the  bath,  and  say  that  the  first  body, 
the  bath,  and  the  thermometer,  have  all  the  same  temperature. 

But  the  body  under  such  circumstances,  as  well  as  the 
bath,  and  the  thermometer,  even  if  they  were  entirely  isolated 
from  external  influences  (which  it  is  convenient  to  suppose 
in  a  theoretical  discussion),  would  be  continually  changing  in 
phase,  and  in  energy  as  well  as  in  other  respects,  although 
our  means  of  observation  are  not  fine  enough  to  perceive 
these  variations. 

The  series  of  phases  through  which  the  whole  system  runs 
in  the  course  of  time  may  not  be  entirely  determined  by  the 
energy,  but  may  depend  on  the  initial  phase  in  other  respects. 
In  such  cases  the  ensemble  obtained  by  the  microcanonical 
distribution  of  the  whole  system,  which  includes  all  possible 
time-ensembles  combined  in  the  proportion  which  seems  least 
arbitrary,  will  represent  better  than  any  one  time-ensemble 
the  effect  of  the  bath.  Indeed  a  single  time-ensemble,  when 
it  is  not  also  a  microcanonical  ensemble,  is  too  ill-defined  a 
notion  to  serve  the  purposes  of  a  general  discussion.  We 
will  therefore  direct  our  attention,  when  we  suppose  the  body 
placed  in  a  bath,  to  the  microcanonical  ensemble  of  phases 
thus  obtained. 

If  we  now  suppose  the  quantity  of  the  substance  forming 
the  bath  to  be  increased,  the  anomalies  of  the  separate  ener- 
gies of  the  body  and  of  the  thermometer  in  the  microcanonical 


THERMODYNAMIC  ANALOGIES.  181 

ensemble  will  be  increased,  but  not  without  limit.  The  anom- 
alies of  the  energy  of  the  bath,  considered  in  comparison  with 
its  whole  energy,  diminish  indefinitely  as  the  quantity  of  the 
bath  is  increased,  and  become  in  a  sense  negligible,  when 
the  quantity  of  the  bath  is  sufficiently  increased.  The 
ensemble  of  phases  of  the  body,  and  of  the  thermometer, 
approach  a  standard  form  as  the  quantity  of  the  bath  is  in- 
definitely increased.  This  limiting  form  is  easily  shown  to  be 
what  we  have  described  as  the  canonical  distribution. 

Let  us  write  e  for  the  energy  of  the  whole  system  consisting 
of  the  body  first  mentioned,  the  bath,  and  the  thermometer 
(if  any),  an4  let  us  first  suppose  this  system  to  be  distributed 
canonically  with  the  modulus  ©.  We  have  by  (205) 


and  since  ep  =  •= 


de  _  n  de 
H®~~2dep' 
If  we  write  Ae  for  the  anomaly  of  mean  square,  we  have 


d® 
If  we  set 


A®  will  represent  approximately  the  increase  of  ®  which 
would  produce  an  increase  in  the  average  value  of  the  energy 
equal  to  its  anomaly  of  mean  square.  Now  these  equations 
give 


(A©)*  =  - 
n 

which  shows  that  we  may  diminish  A  ®  indefinitely  by  increas- 
ing the  quantity  of  the  bath. 

Now  our  canonical  ensemble  consists  of  an  infinity  of  micro- 
canonical  ensembles,  which  differ  only  in  consequence  of  the 
different  values  of  the  energy  which  is  constant  in  each.  If 
we  consider  separately  the  phases  of  the  first  body  which 


182  THERMODYNAMIC  ANALOGIES. 

occur  in  the  canonical  ensemble  of  the  whole  system,  these 
phases  will  form  a  canonical  ensemble  of  the  same  modulus. 
This  canonical  ensemble  of  phases  of  the  first  body  will  con- 
sist of  parts  which  belong  to  the  different  microcanonical 
ensembles  into  which  the  canonical  ensemble  of  the  whole 
system  is  divided. 

Let  us  now  imagine  that  the  modulus  of  the  principal  ca- 
nonical ensemble  is  increased  by  2  A  (8),  and  its  average  energy 
by  2Ae.  The  modulus  of  the  canonical  ensemble  of  the 
phases  of  the  first  body  considered  separately  will  be  increased 
by  2  A  ®.  We  may  regard  the  infinity  of  microcanonical  en- 
sembles into  which  we  have  divided  the  principal  canonical 
ensemble  as  each  having  its  energy  increased  by  2Ae.  Let 
us  see  how  the  ensembles  of  phases  of  the  first  body  con- 
tained in  these  microcanonical  ensembles  are  affected.  We 
may  assume  that  they  will  all  be  affected  in  about  the  same 
way,  as  all  the  differences  which  come  into  account  may  be 
treated  as  small.  Therefore,  the  canonical  ensemble  formed  by 
taking  them  together  will  also  be  affected  in  the  same  way. 
But  we  know  how  this  is  affected.  It  is  by  the  increase  of 
its  modulus  by  2 A®,  a  quantity  which  vanishes  when  the 
quantity  of  the  bath  is  indefinitely  increased. 

In  the  case  of  an  infinite  bath,  therefore,  the  increase  of  the 
energy  of  one  of  the  microcanonical  ensembles  by  2Ae,  pro- 
duces a  vanishing  effect  on  the  distribution  in  energy  of  the 
phases  of  the  first  body  which  it  contains.  But  2Ae  is  more 
than  the  average  difference  of  energy  between  the  micro- 
canonical  ensembles.  The  distribution  in  energy  of  these 
phases  is  therefore  the  same  in  the  different  microcanonical 
ensembles,  and  must  therefore  be  canonical,  like  that  of  the 
ensemble  which  they  form  when  taken  together.* 

*  In  order  to  appreciate  the  above  reasoning,  it  should  be  understood  that 
the  differences  of  energy  which  occur  in  the  canonical  ensemble  of  phases  of 
the  first  body  are  not  here  regarded  as  vanishing  quantities.  To  fix  one's 
ideas,  one  may  imagine  that  he  has  the  fineness  of  perception  to  make  these 
differences  seem  large.  The  difference  between  the  part  of  these  phases 
which  belong  to  one  microcanonical  ensemble  of  the  whole  system  and  the 
part  which  belongs  to  another  would  still  be  imperceptible,  when  the  quan- 
tity of  the  bath  is  sufficiently  increased. 


THERMODYNAMIC  ANALOGIES.  183 

As  a  general  theorem,  the  conclusion  may  be  expressed  in 
the  words :  —  If  a  system  of  a  great  number  of  degrees  of 
freedom  is  microcanonically  distributed  in  phase,  any  very 
small  part  of  it  may  be  regarded  as  canonically  distributed.* 

It  would  seem,  therefore,  that  a  canonical  ensemble  of 
phases  is  what  best  represents,  with  the  precision  necessary 
for  exact  mathematical  reasoning,  the  notion  of  a  body  with 
a  given  temperature,  if  we  conceive  of  the  temperature  as  the 
state  produced  by  such  processes  as  we  actually  use  in  physics 
to  produce  a  given  temperature.  Since  the  anomalies  of  the 
body  increase  with  the  quantity  of  the  bath,  we  can  only  get 
rid  of  all  that  is  arbitrary  in  the  ensemble  of  phases  which  is 
to  represent  the  notion  of  a  body  of  a  given  temperature  by 
making  the  bath  infinite,  which  brings  us  to  the  canonical 
distribution. 

A  comparison  of  temperature  and  entropy  with  their  ana- 
logues in  statistical  mechanics  would  be  incomplete  without  a 
consideration  of  their  differences  with  respect  to  units  and 
zeros,  and  the  numbers  used  for  their  numerical  specification. 
If  we  apply  the  notions  of  statistical  mechanics  to  such  bodies 
as  we  usually  consider  in  thermodynamics,  for  which  the 
kinetic  energy  is  of  the  same  order  of  magnitude  as  the  unit 
of  energy,  but  the  number  of  degrees  of  freedom  is  enormous, 
the  values  of  B,  de/dlogV,  and  de/d<f>  will  be  of  the  same 
order  of  magnitude  as  1/w,  and  the  variable  part  of  ?;,  log  V, 
and  <j>  will  be  of  the  same  order  of  magnitude  as  w.f  If  these 
quantities,  therefore,  represent  in  any  sense  the  notions  of  tem- 
perature and  entropy,  they  will  nevertheless  not  be  measured 
in  units  of  the  usual  order  of  magnitude,  —  a  fact  which  must 
be  borne  in  mind  in  determining  what  magnitudes  may  be 
regarded  as  insensible  to  human  observation. 

Now  nothing  prevents  our  supposing  energy  and  time  in 
our  statistical  formulae  to  be  measured  in  such  units  as  may 

*  It  is-  assumed  —  and  without  this  assumption  the  theorem  would  have 
no  distinct  meaning  —  that  the  part  of  the  ensemble  considered  may  be 
regarded  as  having  separate  energy. 

t  See  equations  (124),  (288),  (289),  and  (314) ;  also  page  106. 


184  THERMODYNAMIC  ANALOGIES. 

be  convenient  for  physical  purposes.  But  when  these  units 
have  been  chosen,  the  numerical  values  of  ®,  de/dlogV, 
de/d<j>,  7),  log  FJ  <£,  are  entirely  determined,*  and  in  order  to 
compare  them  with  temperature  and  entropy,  the  numerical 
values  of  which  depend  upon  an  arbitrary  unit,  we  must  mul- 
tiply all  values  of  ®,  de/dlogV,  de',d^  by  a  constant  (7T), 
and  divide  all  values  of  77,  log  FJ  and  <f>  by  the  same  constant. 
This  constant  is  the  same  for  all  bodies,  and  depends  only  on 
the  units  of  temperature  and  energy  which  we  employ.  For 
ordinary  units  it  is  of  the  same  order  of  magnitude  as  the 
numbers  of  atoms  in  ordinary  bodies. 

We  are  not  able  to  determine  the  numerical  value  of  K> 
as  it  depends  on  the  number  of  molecules  in  the  bodies  with 
which  we  experiment.  To  fix  our  ideas,  however,  we  may 
seek  an  expression  for  this  value,  based  upon  very  probable 
assumptions,  which  will  show  how  we  would  naturally  pro- 
ceed to  its  evaluation,  if  our  powers  of  observation  were  fine 
enough  to  take  cognizance  of  individual  molecules. 

If  the  unit  of  mass  of  a  monatomic  gas  contains  v  atoms, 
and  it  may  be  treated  as  a  system  of  3  v  degrees  of  free- 
dom, which  seems  to  be  the  case,  we  have  for  canonical 
distribution 


If  we  write  T  for  temperature,  and  cv  for  the  specific  heat-  of 
the  gas  for  constant  volume  (or  rather  the  limit  toward 
which  this  specific  heat  tends,  as  rarefaction  is  indefinitely 
increased),  we  have 


since  we  may  regard  the  energy  as  entirely  kinetic.     We  may 
set  the  ep  of  this  equation  equal  to  the  ep  of  the  preceding, 

*  The  unit  of  time  only  affects  the  last  three  quantities,  and  these  only 
by  an  additive  constant,  which  disappears  (with  the  additive  constant  of 
entropy),  when  differences  of  entropy  are  compared  with  their  statistical 
analogues.  See  page  19. 


THERMODYNAMIC  ANALOGIES.  185 

where  indeed  the  individual  values  of  which  the  average  is 
taken  would  appear  to  human  observation  as  identical.     This 

gives 

d®       2cv 


whence  ='  <493) 

a  value  recognized  by  physicists  as  a  constant  independent  of 
the  kind  of  monatomic  gas  considered. 

We  may  also  express  the  value  of  K  in  a  somewhat  different 
form,  which  corresponds  to  the  indirect  method  by  which 
physicists  are  accustomed  to  determine  the  quantity  cv.  The 
kinetic  energy  due  to  the  motions  of  the  centers  of  mass  of 
the  molecules  of  a  mass  of  gas  sufficiently  expanded  is  easily 
shown  to  be  equal  to 


where  p  and  v  denote  the  pressure  and  volume.  The  average 
value  of  the  same  energy  in  a  canonical  ensemble  of  such 
a  mass  of  gas  is 

J0v, 

where  v  denotes  the  number  of  molecules  in  the  gas.  Equat- 
ing these  values,  we  have 

pv  =  ®v,  (494) 

whence  J£~~T~^'  (495) 

Now  the  laws  of  Boyle,  Charles,  and  Avogadro  may  be  ex- 
pressed by  the  equation 

pv  —  AvT,  (496) 

where  A  is  a  constant  depending  only  on  the  units  hi  which 
energy  and  temperature  are  measured.  1  /  K,  therefore,  might 
be  called  the  constant  of  the  law  of  Boyle,  Charles,  and 
Avogadro  as  expressed  with  reference  to  the  true  number  of 
molecules  in  a  gaseous  body. 

Since  such  numbers  are  unknown  to  us,  it  is  more  conven- 
ient to  express  the  law  with  reference  to  relative  values.  If 
we  denote  by  M  the  so-called  molecular  weight  of  a  gas,  that 


186  THERMODYNAMIC  ANALOGIES. 

is,  a  number  taken  from  a  table  of  numbers  proportional  to 
the  weights  of  various  molecules  and  atoms,  but  having  one 
of  the  values,  perhaps  the  atomic  weight  of  hydrogen,  arbi- 
trarily made  unity,  the  law  of  Boyle,  Charles,  and  Avogadro 
may  be  written  in  the  more  practical  form 

pv  =  A'T-^,  (497) 

JXL 

where  A'  is  a  constant  and  m  the  weight  of  gas  considered. 
It  is  evident  that  1  K  is  equal  to  the  product  of  the  constant 
of  the  law  in  this  form  and  the  (true)  weight  of  an  atom  of 
hydrogen,  or  such  other  atom  or  molecule  as  may  be  given 
the  value  unity  in  the  table  of  molecular  weights. 

In  the  following  chapter  we  shall  consider  the  necessary 
modifications  in  the  theory  of  equilibrium,  when  the  quantity 
of  matter  contained  in  a  system  is  to  be  regarded  as  variable, 
or,  if  the  system  contains  more  than  one  kind  of  matter, 
when  the  quantities  of  the  several  kinds  of  matter  in  the 
system  are  to  be  regarded  as  independently  variable.  This 
will  give  us  yet  another  set  of  variables  in  the  statistical 
equation,  corresponding  to  those  of  the  amplified  form  of 
the  thennodynamic  equation. 


CHAPTER  XV. 

SYSTEMS   COMPOSED  OF  MOLECULES. 

THE  nature  of  material  bodies  is  such,  that  especial  interest 
attaches  to  the  dynamics  of  systems  composed  of  a  great 
number  of  entirely  similar  particles,  or,  it  may  be,  of  a  great 
number  of  particles  of  several  kinds,  all  of  each  kind  being 
entirely  similar  to  each  other.  We  shall  therefore  proceed  to 
consider  systems  composed  of  such  particles,  whether  in  great 
numbers  or  otherwise,  and  especially  to  consider  the  statistical 
equilibrium  of  ensembles  of  such  systems.  One  of  the  varia- 
tions to  be  considered  in  regard  to  such  systems  is  a  variation 
in  the  numbers  of  the  particles  of  the  various  kinds  which  it 
contains,  and  the  question  of  statistical  equilibrium  between 
two  ensembles  of  such  systems  relates  in  part  to  the  tendencies 
of  the  various  kinds  of  particles  to  pass  from  the  one  to  the 
other. 

First  of  all,  we  must  define  precisely  what  is  meant  by 
statistical  equilibrium  of  such  an  ensemble  of  systems.  The 
essence  of  statistical  equilibrium  is  the  permanence  of  the 
number  of  systems  which  fall  within  any  given  limits  with 
respect  to  phase.  We  have  therefore  to  define  how  the  term 
"  phase  "  is  to  be  understood  in  such  cases.  If  two  phases  differ 
only  in  that  certain  entirely  similar  particles  have  changed 
places  with  one  another,  are  they  to  be  regarded  as  identical 
or  different  phases?  If  the  particles  are  regarded  as  indis- 
tinguishable, it  seems  in  accordance  with  the  spirit  of  the 
statistical  method  to  regard  the  phases  as  identical.  In  fact, 
it  might  be  urged  that  in  such  an  ensemble  of  systems  as  we 
are  considering  no  identity  is  possible  between  the  particles 
of  different  systems  except  that  of  qualities,  and  if  v  particles 
of  one  system  are  described  as  entirely  similar  to  one  another 
and  to  v  of  another  system,  nothing  remains  on  which  to  base 


188  SYSTEMS   COMPOSED   OF  MOLECULES. 

the  indentification  of  any  particular  particle  of  the  first  system 
with  any  particular  particle  of  the  second.  And  this  would 
be  true,  if  the  ensemble  of  systems  had  a  simultaneous 
objective  existence.  But  it  hardly  applies  to  the  creations 
of  the  imagination.  In  the  cases  which  we  have  been  con- 
sidering, and  in  those  which  we  shall  consider,  it  is  not  only 
possible  to  conceive  of  the  motion  of  an  ensemble  of  similar 
systems  simply  as  possible  cases  of  the  motion  of  a  single 
system,  but  it  is  actually  in  large  measure  for  the  sake  of 
representing  more  clearly  the  possible  cases  of  the  motion  of 
a  single  system  that  we  use  the  conception  of  an  ensemble 
of  systems.  The  perfect  similarity  of  several  particles  of  a 
system  will  not  in  the  least  interfere  with  the  identification 
of  a  particular  particle  in  one  case  with  a  particular  particle 
in  another.  The  question  is  one  to  be  decided  in  accordance 
with  the  requirements  of  practical  convenience  in  the  discus- 
sion of  the  problems  with  which  we  are  engaged. 

Our  present  purpose  will  often  require  us  to  use  the  terms 
phase,  density-in-phase,  statistical  equilibrium,  and  other  con- 
nected terms  on  the  supposition  that  phases  are  not  altered 
by  the  exchange  of  places  between  similar  particles.  Some 
of  the  most  important  questions  with  which  we  are  concerned 
have  reference  to  phases  thus  defined.  We  shall  call  them 
phases  determined  by  generic  definitions,  or  briefly,  generic 
phases.  But  we  shall  also  be  obliged  to  discuss  phases  de- 
fined by  the  narrower  definition  (so  that  exchange  of  position 
between  similar  particles  is  regarded  as  changing  the  phase), 
which  will  be  called  phases  determined  by  specific  definitions, 
or  briefly,  specific  phases.  For  the  analytical  description  of 
a  specific  phase  is  more  simple  than  that  of  a  generic  phase. 
And  it  is  a  more  simple  matter  to  make  a  multiple  integral 
extend  over  all  possible  specific  phases  than  to  make  one  extend 
without  repetition  over  all  possible  generic  phases. 

It  is  evident  that  if  i>i,  vz  .  .  .  vh,  are  the  numbers  of  the  dif- 
ferent kinds  of  molecules  in  any  system,  the  number  of  specific 
phases  embraced  in  one  generic  phase  is  represented  by  the 
continued  product  [z^  [^  •  •  •  ]^  and  the  coefficient  of  probabil- 


SYSTEMS   COMPOSED  OF  MOLECULES.  189 

ity  of  a  generic  phase  is  the  sum  of  the  probability-coefficients 
of  the  specific  phases  which  it  represents.  When  these  are 
equal  among  themselves,  the  probability-coefficient  of  the  gen- 
eric phase  is  equal  to  that  of  the  specific  phase  multiplied  by 
[z/i  1 1>2  .  .  .  \vg  It  is  also  evident  that  statistical  equilibrium 
may  subsist  with  respect  to  generic  phases  without  statistical 
equilibrium  with  respect  to  specific  phases,  but  not  vice  versa. 

Similar  questions  arise  where  one  particle  is  capable  of 
several  equivalent  positions.  Does  the  change  from  one  of 
these  positions  to  another  change  the  phase?  It  would  be 
most  natural  and  logical  to  make  it  affect  the  specific  phase, 
but  not  the  generic.  The  number  of  specific  phases  contained 
in  a  generic  phase  would  then  be  \v±  /e/1  .  .  .  |z^  /ch\  where 
KV  .  .  .  Kh  denote  the  numbers  of  equivalent  positions  belong- 
ing to  the  several  kinds  of  particles.  The  case  in  which  a  K  is 
infinite  would  then  require  especial  attention.  It  does  not 
appear  that  the  resulting  complications  in  the  formulae  would 
be  compensated  by  any  real  advantage.  The  reason  of  this  is 
that  in  problems  of  real  interest  equivalent  positions  of  a 
particle  will  always  be  equally  probable.  In  this  respect, 
equivalent  positions  of  the  same  particle  are  entirely  unlike 
the  [^different  ways  in  which  v  particles  may  be  distributed 
in  v  different  positions.  Let  it  therefore  be  understood  that 
in  spite  of  the  physical  equivalence  of  different  positions  of 
the  same  particle  they  are  to  be  considered  as  constituting  a 
difference  of  generic  phase  as  well  as  of  specific.  The  number 
of  specific  phases  contained  in  a  generic  phase  is  therefore 
always  given  by  the  product  \v^\v^  •  »  .  [iy 

Instead  of  considering,  as  in  the  preceding  chapters,  en- 
sembles of  systems  differing  only  in  phase,  we  shall  now 
suppose  that  the  systems  constituting  an  ensemble  are  com- 
posed of  particles  of  various  kinds,  and  that  they  differ  not 
only  in  phase  but  also  in  the  numbers  of  these  particles  which 
they  contain.  The  external  coordinates  of  all  the  systems  in 
the  ensemble  are  supposed,  as  heretofore,  to  have  the  same 
value,  and  when  they  vary,  to  vary  together.  For  distinction, 
we  may  call  such  an  ensemble  a  grand  ensemble,  and  one  in 


190  SYSTEMS  COMPOSED   OF  MOLECULES. 

which  the  systems  differ  only  in  phase  a  petit  ensemble.  A 
grand  ensemble  is  therefore  composed  of  a  multitude  of  petit 
ensembles.  The  ensembles  which  we  have  hitherto  discussed 
are  petit  ensembles. 

Let  i>j,  .  .  .  vh9  etc.,  denote  the  numbers  of  the  different 
kinds  of  particles  in  a  system,  e  its  energy,  and  ql1  .  .  .  qn, 
pl  ,  .  .  .  pn  its  coordinates  and  momenta.  If  the  particles  are  of 
the  nature  of  material  points,  the  number  of  coordinates  (n) 
of  the  system  will  be  equal  to  3  vl  .  .  .  +  3  vh.  But  if  the  parti- 
cles are  less  simple  in  their  nature,  if  they  are  to  be  treated 
as  rigid  solids,  the  orientation  of  which  must  be  regarded,  or 
if  they  consist  each  of  several  atoms,  so  as  to  have  more  than 
three  degrees  of  freedom,  the  number  of  coordinates  of  the 
system  will  be  equal  to  the  sum  of  vlt  i>2,  etc.,  multiplied 
each  by  the  number  of  degrees  of  freedom  of  the  kind  of 
particle  to  which  it  relates. 

Let  us  consider  an  ensemble  in  which  the  number  of 
systems  having  v19  .  .  .  vh  particles  of  the  several  kinds,  and 
having  values  of  their  coordinates  and  momenta  lying  between 
the  limits  ql  and  q^  +  dq1  ,  p1  and  pl  +  dpl  ,  etc.,  is  represented 
by  the  expression 


(498) 


where  IV,  O,  ®,  /^  ,  .  .  .  ph  are  constants,  N  denoting  the  total 
number  of  systems  in  the  ensemble.     The  expression 

Q-f  Wi  - 


Ne  ®  (499) 

[vi."h 

evidently  represents  the  density-in-phase  of  the  ensemble 
within  the  limits  described,  that  is,  for  a  phase  specifically 
defined.  The  expression 


e  *  (500) 


SYSTEMS  COMPOSED  OF  MOLECULES.  191 

is  therefore  the  probability-coefficient  for  a  phase  specifically 
defined.  This  has  evidently  the  same  value  for  all  the 
[iY  .  .  .  \vh  phases  obtained  by  interchanging  the  phases  of 
particles  of  the  same  kind.  The  probability-coefficient  for  a 
generic  phase  will  be  \vi_.  .  .  [z^  times  as  great,  viz., 


e  .  (501) 

We  shall  say  that  such  an  ensemble  as  has  been  described 
is  canonically  distributed,  and  shall  call  the  constant  ©  its 
modulus.  It  is  evidently  what  we  have  called  a  grand  ensem- 
ble. The  petit  ensembles  of  which  it  is  composed  are 
canonically  distributed,  according  to  the  definitions  of  Chapter 
IV,  since  the  expression 


(502) 


is  constant  for  each  petit  ensemble.  The  grand  ensemble, 
therefore,  is  in  statistical  equilibrium  with  respect  to  specific 
phases. 

If  an  ensemble,  whether  grand  or  petit,  is  identical  so  far 
as  generic  phases  are  concerned  with  one  canonically  distrib- 
uted, we  shall  say  that  its  distribution  is  canonical  with 
respect  to  generic  phases.  Such  an  ensemble  is  evidently  in 
statistical  equilibrium  with  respect  to  generic  phases,  although 
it  may  not  be  so  with  respect  to  specific  phases. 

If  we  write  H  for  the  index  of  probability  of  a  generic  phase 
in  a  grand  ensemble,  we  have  for  the  case  of  canonical 
distribution 

H  =  0  +  M.n  —  +  >*»*-«  _  (503) 

It  will  be  observed  that  the  H  is  a  linear  function  of  e  and 
vv  .  .  .  vh  ;  also  that  whenever  the  index  of  probability  of 
generic  phases  in  a  grand  ensemble  is  a  linear  function  of 
e,  j/j,  .  .  .  vhi  the  ensemble  is  canonically  distributed  with 
respect  to  generic  phases. 


192  SYSTEMS   COMPOSED   OF  MOLECULES. 

The   constant  Ii   we   may   regard  as  determined  by   the 
equation 


/C]\TP         ® 
•     /  ^n  -  i  -  dp,...  dqn,        (504) 

phases  J  ln-'-b_ 

or 


[1/1  .  .  .  [ 

'  —    phases 


(505) 


where  the  multiple  sum  indicated  by  2Vl  .  .  .  2rft  includes  all 
terms  obtained  by  giving  to  each  of  the  symbols  vi  .  .  .  vh  all 
integral  values  from  zero  upward,  and  the  multiple  integral 
(which  is  to  be  evaluated  separately  for  each  term  of  the 
multiple  sum)  is  to  be  extended  over  all  the  (specific)  phases 
of  the  system  having  the  specified  numbers  of  particles  of  the 
various  kinds.  The  multiple  integral  hi  the  last  equation  is 

JL 

what  we  have  represented  by  e  0.  See  equation  (92).  We 
may  therefore  write 


It  should  be  observed  that  the  summation  includes  a  term 
in  which  all  the  symbols  vl  .  .  .  vh  have  the  value  zero.  We 
must  therefore  recognize  in  a  certain  sense  a  system  consisting 
of  no  particles,  which,  although  a  barren  subject  of  study  in 
itself,  cannot  well  be  excluded  as  a  particular  case  of  a  system 
of  a  variable  number  of  particles.  In  this  case  e  is  constant, 
and  there  are  no  integrations  to  be  performed.  We  have 
therefore* 

_4       _1 
e   ®  =  e   ®,     i.  e.y     \j/  =  e. 

*  This  conclusion  may  appear  a  little  strained.  The  original  definition 
of  ^  may  not  be  regarded  as  fairly  applying  to  systems  of  no  degrees  of 
freedom.  We  may  therefore  prefer  to  regard  these  equations  as  defining 
4/  in  this  case. 


SYSTEMS  COMPOSED  OF  MOLECULES.  193 

The  value  of  ep  is  of  course  zero  in  this  case.  But  the 
value  of  eq  contains  an  arbitrary  constant,  which  is  generally 
determined  by  considerations  of  convenience,  so  that  eg  and  e 
do  not  necessarily  vanish  with  v^ ,  .  .  .  vh. 

Unless  —  II  has  a  finite  value,  our  formulae  become  illusory. 
We  have  already,  in  considering  petit  ensembles  canonically 
distributed,  found  it  necessary  to  exclude  cases  in  which  —  ty 
has  not  a  finite  value.*  The  same  exclusion  would  here 
make  —  ^r  finite  for  any  finite  values  of  vl  .  .  .  vh.  This  does 
not  necessarily  make  a  multiple  series  of  the  form  (506)  finite. 
We  may  observe,  however,  that  if  for  all  values  of  vl  .  .  .  vh 

\l/    ^.    CQ   +    ^1   Vl)    •    •    •    4"    Ch  Vht  (507) 

where  £0,  cv  .  .  .  ch  are  constants  or  functions  of  ®, 

Co-MMl+CjK  .  .  .  -K/*A+CAVA 


e 


^ 


_n       £p 
&       e 


O.          c_0 
0^0 


.  .  .  e 


-£+•    e   .-.  +  «     e  •  (508) 

The  value  of  —  II  will  therefore  be  finite,  when  the  condition 
(507)  is  satisfied.  If  therefore  we  assume  that  —  fl  is  finite, 
we  do  not  appear  to  exclude  any  cases  which  are  analogous  to 
those  of  nature.f 

The  interest  of  the  ensemble  which  has  been  described  lies 
in  the  fact  that  it  may  be  in  statistical  equilbrium,  both  in 

*  See  Chapter  IV,  page  35. 

t  If  the  external  coordinates  determine  a  certain  volume  within  which  the 
system  is"  confined,  the  contrary  of  (507)  would  imply  that  we  could  obtain 
an  infinite  amount  of  work  by  crowding  an  infinite  quantity  of  matter  into  a 
finite  volume. 

13 


194  SYSTEMS   COMPOSED  OF  MOLECULES. 

respect  to  exchange  of  energy  and  exchange  of  particles,  with 
other  grand  ensembles  canonically  distributed  and  having  the 
same  values  of  ®  and  of  the  coefficients  pv  ^2,  etc.,  when  the 
circumstances  are  such  that  exchange  of  energy  and  of 
particles  are  possible,  and  when  equilibrium  would  not  sub- 
sist, were  it  not  for  equal  values  of  these  constants  in  the  two 
ensembles. 

With  respect  to  the  exchange  of  energy,  the  case  is  exactly 
the  same  as  that  of  the  petit  ensembles  considered  in  Chapter 
IV,  and  needs  no  especial  discussion.  The  question  of  ex- 
change of  particles  is  to  a  certain  extent  analogous,  and  may 
be  treated  in  a  somewhat  similar  manner.  Let  us  suppose 
that  we  have  two  grand  ensembles  canonically  distributed 
with  respect  to  specific  phases,  with  the  same  value  of  the 
modulus  and  of  the  coefficients  ^  .  .  .  fih  ,  and  let  us  consider 
the  ensemble  of  all  the  systems  obtained  by  combining  each 
system  of  the  first  ensemble  with  each  of  the  second. 

The  probability-coefficient  of  a  generic  phase  in  the  first 
ensemble  may  be  expressed  by 


e  &  (509) 

The  probability-coefficient  of  a  specific  phase  will  then  be 
expressed  by 


(510) 


since  each  generic  phase  comprises  \v^ .  .  .  [z^  specific  phases. 
In  the  second  ensemble  the  probability-coefficients  of  the 
generic  and  specific  phases  will  be 


SYSTEMS   COMPOSED  OF  MOLECULES.  195 

The  probability-coefficient  of  a  generic  phase  in  the  third 
ensemble,  which  consists  of  systems  obtained  by  regarding 
each  system  of  the  first  ensemble  combined  with  each  of  the 
second  as  forming  a  system,  will  be  the  product  of  the  proba- 
bility-coefficients of  the  generic  phases  of  the  systems  com- 
bined, and  will  therefore  be  represented  by  the  formula 


e  (513) 

where  ft"'  =  ft'  +  ft",  e'"  =  e'  +  e",  vi'"  =  vj  +  z>i",  etc.  It 
will  be  observed  that  i//",  etc.,  represent  the  numbers  of 
particles  of  the  various  kinds  in  the  third  ensemble,  and  e'" 
its  energy  ;  also  that  ft'"  is  a  constant.  The  third  ensemble 
is  therefore  canonically  distributed  with  respect  to  generic 
phases. 

If  all  the  systems  in  the  same  generic  phase  in  the  third 
ensemble  were  equably  distributed  among  the  zV"  •  •  •  |  vjj"  spe- 


cific phases  which  are  comprised  in  the  generic  phase,  the  prob- 
ability-coefficient of  a  specific  phase  would  be 


In  fact,  however,  the   probability-coefficient  of  any  specific 
phase  which  occurs  in  the  third  ensemble  is 


which  we  get  by  multiplying  the  probability-coefficients  of 
specific  phases  in  the  first  and  second  ensembles.  The  differ- 
ence between  the  formulae  (514)  and  (515)  is  due  to  the  fact 
that  the  generic  phases  to  which  (513)  relates  include  not 
only  the  specific  phases  occurring  in  the  third  ensemble  and 
having  the  probability-coefficient  (515),  but  also  all  the 
specifier  phases  obtained  from  these  by  interchange  of  similar 
particles  between  two  combined  systems.  Of  these  the  proba- 


196  SYSTEMS   COMPOSED  OF  MOLECULES. 

bility-coefficient  is  evidently  zero,  as  they  do  not  occur  in  the 
ensemble. 

Now  this  third  ensemble  is  in  statistical  equilibrium,  with 
respect  both  to  specific  and  generic  phases,  since  the  ensembles 
from  which  it  is  formed  are  so.  This  statistical  equilibrium 
is  not  dependent  on  the  equality  of  the  modulus  and  the  co-effi- 
cients /Aj ,  .  .  .  fxh  in  the  first  and  second  ensembles.  It  depends 
only  on  the  fact  that  the  two  original  ensembles  were  separ- 
ately in  statistical  equilibrium,  and  that  there  is  no  interaction 
between  them,  the  combining  of  the  two  ensembles  to  form  a 
third  being  purely  nominal,  and  involving  no  physical  connec- 
tion. This  independence  of  the  systems,  determined  physically 
by  forces  which  prevent  particles  from  passing  from  one  sys- 
tem to  the  other,  or  coming  within  range  of  each  other's  action, 
is  represented  mathematically  by  infinite  values  of  the  energy 
for  particles  in  a  space  dividing  the  systems.  Such  a  space 
may  be  called  a  diaphragm. 

If  we  now  suppose  that,  when  we  combine  the  systems  of 
the  two  original  ensembles,  the  forces  are  so  modified  that  the 
energy  is  nc  longer  infinite  for  particles  in  all  the  space  form- 
ing the  diaphragm,  but  is  diminished  in  a  part  of  this  space, 
so  that  it  is  possible  for  particles  to  pass  from  one  system 
to  the  other,  this  will  involve  a  change  in  the  function  e;// 
which  represents  the  energy  of  the  combined  systems,  and  the 
equation  e"f  —  ef  +  eff  will  no  longer  hold.  Now  if  the  co- 
efficient of  probability  in  the  third  ensemble  were  represented 
by  (513)  with  this  new  function  e;//,  we  should  have  statistical 
equilibrium,  with  respect  to  generic  phases,  although  not  to 
specific.  But  this  need  involve  only  a  trifling  change  in  the 
distribution  of  the  third  ensemble,*  a  change  represented  by 
the  addition  of  comparatively  few  systems  in  which  the  trans- 
ference of  particles  is  taking  place  to  the  immense  number 

*  It  will  be  observed  that,  so  far  as  the  distribution  is  concerned,  very 
large  and  infinite  values  of  e  (for  certain  phases)  amount  to  nearly  the  same 
thing,  —  one  representing  the  total  and  the  other  the  nearly  total  exclusion 
of  the  phases  in  question.  An  infinite  change,  therefore,  in  the  value  of  e 
(for  certain  phases)  may  represent  a  vanishing  change  in  the  distribution. 


SYSTEMS   COMPOSED  OF  MOLECULES.  197 

obtained  by  combining  the  two  original  ensembles.  The 
difference  between  the  ensemble  which  would  be  in  statistical 
equilibrium,  and  that  obtained  by  combining  the  two  original 
ensembles  may  be  diminished  without  limit,  while  it  is  still 
possible  for  particles  to  pass  from  one  system  to  another.  In  - 
this  sense  we  may  say  that  the  ensemble  formed  by  combining 
the  two  given  ensembles  may  still  be  regarded  as  in  a  state  of 
(approximate)  statistical  equilibrium  with  respect  to  generic 
phases,  when  it  has  been  made  possible  for  particles  to  pass 
between  the  systems  combined,  and  when  statistical  equilibrium 
for  specific  phases  has  therefore  entirely  ceased  to  exist,  and 
when  the  equilibrium  for  generic  phases  would  also  have 
entirely  ceased  to  exist,  if  the  given  ensembles  had  not  been 
canonically  distributed,  with  respect  to  generic  phases,  with 
the  same  values  of  @  and  fiv  .  .  .  ph. 

It  is  evident  also  that  considerations  of  this  kind  will  apply  j 
separately  to  the  several  kinds  of  particles.     We  may  diminish  ' 
the  energy  in  the  space  forming  the  diaphragm  for  one  kind  of 
particle  and  not  for  another.     This  is  the  mathematical  ex- 
pression for  a  "  semipermeable"  diaphragm.      The  condition 
necessary  for  statistical  equilibrium  where  the  diaphragm  is 
permeable  only  to  particles  to  which  the  suffix  (  )x  relates 
will  be  fulfilled  when  /^  and  ®  have  the  same  values  in  the 
two  ensembles,  although  the  other  coefficients  /*2,  etc.,  may  be 
different. 

This  important  property  of  grand  ensembles  with  canonical 
distribution  will  supply  the  motive  for  a  more  particular  ex- 
amination of  the  nature  of  such  ensembles,  and  especially  of 
the  comparative  numbers  of  systems  in  the  several  petit  en- 
sembles which  make  up  a  grand  ensemble,  and  of  the  average 
values  in  the  grand  ensemble  of  some  of  the  most  important 
quantities,  and  of  the  average  squares  of  the  deviations  from 
these  average  values. 

The  probability  that  a  system  taken  at  random  from  a 
grand  ensemble  canonically  distributed  will  have  exactly 
i/j,  .  .  .  vh  particles  of  the  various  kinds  is  expressed  by  the 
multiple  integral 


198  SYSTEMS  COMPOSED  OF  MOLECULES. 


phases 


or  «  .  (517) 

[vi  .  .  .  [1/5 

This  may  be  called  the  probability  of  the  petit  ensemble 
0>i,  ...  vh).  The  sum  of  all  such  probabilities  is  evidently 
unity.  That  is, 


(518) 


which  agrees  with  (506). 

The  average  value  in  the  grand  ensemble  of  any  quantity 
u,  is  given  by  the  formula 


phases 

If  w  is  a  function  of  i/x,  .  .  .  i/A  alone,  i.  e.,  if  it  has  the  same 
value  in  all  systems  of  any  same  petit  ensemble,  the  formula 
reduces  to 


8   ue 

=  X"' 


Again,  if  we  write  ^grand  an(i  w]  petit  to  distinguish  averages  in 
the  grand  and  petit  ensembles,  we  shall  have 


In  this  chapter,  in  which  we  are  treating  of  grand  en- 
sembles, u  will  always  denote  the  average  for  a  grand  en- 
semble. In  the  preceding  chapters,  u  has  always  denoted 
the  average  for  a  petit  ensemble. 


SYSTEMS   COMPOSED  OF  MOLECULES.  199 

Equation  (505),  which  we  repeat  in  a   slightly  different 
form,  viz., 


phases 


shows  that  O  is  a  function  of  ®  and  pv  .  .  .  fj,h  ;  also  of  the 
external  coordinates  «1?  a2,  etc.,  which  are  involved  implicitly 
in  e.  If  we  differentiate  the  equation  regarding  all  these 
quantities  as  variable,  we  have 


la 

phases 


phases 

+  etc. 


all 

de  e 


phases 

-  etc.  (523) 

5 

If  we  multiply  this  equation  by  e9,  and  set  as  usual  Av  Av 
etc.,  for  —  de/da^  —  delda^,  etc.,  we  get  in  virtue  of  the  law 
expressed  by  equation  (519), 

dto      O   _  d®        - 


200  SYSTEMS   COMPOSED   OF  MOLECULES. 

that  is, 

da  =  O  +  PI*  —  ft*-?  ^  _  a  -  fa  _  s  2i  dai      (525) 

Since  equation  (503)  gives 


the  preceding  equation  may  be  written 

dQ,  —  ILd®  —  2  vid/xi  —  2  2l  dalt  (527) 
Again,  equation  (526)  gives 

c?Q  +  Sjiie^i  +  S  vi  cZ/*!  —  de  =  ©dH  +  Hc2®.  (528) 
Eliminating  <#fl  from  these  equations,  we  get 

de  =  —  ©rfH  +  2/x^i  -  S^j  rfoj.  (529) 

If  we  set               *  =  e  +  ©  H,  (530) 

d*  =  de  +  ©  dH  +  H  d®,  (531) 

we  have             d*  =  H  d®  +  2  ^  d^  -  S  -^  e?^.  (532) 

The  corresponding  thermodynamic  equations  are 

de  =  Tdy  +  5  ^dmi  —  S  -4i  ^  ,  (533) 

(534) 

/xi  cZm!  —  SA  <?%  .  (535) 


These  are  derived  from  the  thermodynamic  equations  (114) 
and  (11  7)  by  the  addition  of  the  terms  necessary  to  take  ac- 
count of  variation  in  the  quantities  (mv  mv  etc.)  of  the 
several  substances  of  which  a  body  is  composed.  The  cor- 
respondence of  the  equations  is  most  perfect  when  the  com- 
ponent substances  are  measured  in  such  units  that  mv  m2, 
etc.,  are  proportional  to  the  numbers  of  the  different  kinds 
of  molecules  or  atoms.  The  quantities  pv  p2,  etc.,  in  these 
thermodynamic  equations  may  be  defined  as  differential  coeffi- 
cients by  either  of  the  equations  in  which  they  occur.* 

*  Compare  Transactions  Connecticut  Academy,  Vol.  Ill,  pages  116  ff. 


SYSTEMS   COMPOSED   OF  MOLECULES.  201 

If  we  compare  the  statistical  equations  (529)  and  (532) 
with  (114)  and  (112),  which  are  given  in  Chapter  IV,  and 
discussed  in  Chapter  XIV,  as  analogues  of  thermody- 
namic  equations,  we  find  considerable  difference.  Beside  the 
terms  corresponding  to  the  additional  terms  in  the  thermo- 
dynamic  equations  of  this  chapter,  and  beside  the  fact  that 
the  averages  are  taken  in  a  grand  ensemble  in  one  case 
and  in  a  petit  in  the  other,  the  analogues  of  entropy,  H 
and  ?/,  are  quite  different  in  definition  and  value.  We  shall 
return  to  this  point  after  we  have  determined  the  order 
of  magnitude  of  the  usual  anomalies  of  vv  ...  vh. 

If  we  differentiate  equation  (518)  with  respect  to  /ii,  and 
multiply  by  <H),  we  get 


"0*   h  —  b.  "  =  0?        (536) 

whence  dtl/d^  =  —  vv  which  agrees  with  (527).     Differen- 
tiating again  with  respect  to  tav  and  to  /*2,  and  setting 

we  get 

-     _«,  _  to.  .  ^  rf^% 


|  vi 


=  0)       (537) 


The  first  members  of  these  equations  represent  the  average 
values  of  the  quantities  in  the  principal  parentheses.  We 
have  therefore 


,  =  ©  ^  ,  (539) 

=  ®^r  =  ®lr-  (54°) 


202  SYSTEMS   COMPOSED  OF  MOLECULES. 

From  equation  (539)  we  may  get  an  idea  of  the  order  of 
magnitude  of  the  divergences  of  vl  from  its  average  value 
in  the  ensemble,  when  that  average  value  is  great.  The 
equation  may  be  written 


(541) 


The  second  member  of  this  equation  will  in  general  be  small 
when  j/j  is  great.  Large  values  are  not  necessarily  excluded, 
but  they  must  be  confined  within  very  small  limits  with  re- 
spect to  /JL.  For  if 


(542) 


for  all  values  of  ^i  between  the  limits  /*/  and  /-tj",  we  shall 
have  between  the  same  limits 

—  dvi  >  cZ/xi ,  (543) 

and  therefore 

/I         1  \ 

>  m"  -  Hi'-  (544) 


The  difference  /*/'  —  ^  is  therefore  numerically  a  very  small 
quantity.  To  form  an  idea  of  the  importance  of  such  a 
difference,  we  should  observe  that  in  formula  (498)  ^  is 
multiplied  by  v1  and  the  product  subtracted  from  the  energy. 
A  very  small  difference  in  the  value  of  /^  may  therefore  be  im- 
portant. But  since  v  <B)  is  always  less  than  the  kinetic  energy 
of  the  system,  our  formula  shows  that  ^'  —  //,/,  even  when 
multiplied  by  vj  or  i^",  may  still  be  regarded  as  an  insensible 
quantity. 

We  can  now  perceive  the  leading  characteristics  with  re- 
spect to  properties  sensible  to  human  faculties  of  such  an  en- 
semble as  we  are  considering  (a  grand  ensemble  canonically 
distributed),  when  the  average  numbers  of  particles  of  the  vari- 
ous kinds  are  of  the  same  order  of  magnitude  as  the  number 
of  molecules  in  the  bodies  which  are  the  subject  of  physical 


SYSTEMS   COMPOSED  OF  MOLECULES.  203 

experiment.  Although  the  ensemble  contains  systems  having 
the  widest  possible  variations  in  respect  to  the  numbers  of 
the  particles  which  they  contain,  these  variations  are  practi- 
cally contained  within  such  narrow  limits  as  to  be  insensible, 
except  for  particular  values  of  the  constants  of  the  ensemble. 
This  exception  corresponds  precisely  to  the  case  of  nature, 
when  certain  therm  odynamic  quantities  corresponding  to  ®, 
/ii,  /A2,  etc.,  which  in  general  determine  the  separate  densities 
of  various  components  of  a  body,  have  certain  values  which 
make  these  densities  indeterminate,  in  other  words,  when  the 
conditions  are  such  as  determine  coexistent  phases  of  matter. 
Except  in  the  case  of  these  particular  values,  the  grand  en- 
semble would  not  differ  to  human  faculties  of  perception  from 
a  petit  ensemble,  viz.,  any  one  of  the  petit  ensembles  which  it 
contains  in  which  j^,  j/2,  etc.,  do  not  sensibly  differ  from  their 
average  values. 

Let  us  now  compare  the  quantities  H  and  77,  the  average 
values  of  which  (in  a  grand  and  a  petit  ensemble  respectively) 
we  have  seen  to  correspond  to  entropy.  Since 

__ 


,  \b  —  € 

and  *= 


(545) 


A  part  of  this  difference  is  due  to  the  fact  that  H  relates  to 
generic  phases  and  77  to  specific.  If  we  write  ?7gen  for  the 
index  of  probability  for  generic  phases  in  a  petit  ensemble, 
we  have 

^gen  =  ?!  +  ^g  [Vl  .   .  .  [vfc  ,  (546) 

H  -  r,  =  H  -  ^  +  log  |vi  .  .  .  [v»  ,  (547) 

.!^      (548) 


This   is   the   logarithm   of  the   probability  of   the  petit  en- 
semble (vl  .  .  .  vh)*     If  we  set 

*  See  formula  (517). 


204  SYSTEMS   COMPOSED   OF  MOLECULES. 

-^ =  >7gen>  (549) 

which  corresponds  to  the  equation 


we  have  i/^  =  $  +  ®  log  [v, 

and  H-^^n  +  ^yi--@+°.  (551) 

This  will  have  a  maximum  when  * 


Distinguishing  values  corresponding  to  this  maximum  by 
accents,  we  have  approximately,  when  vl  ,  .  .  .  vh  are  of  the 
same  order  of  magnitude  as  the  numbers  of  molecules  in  ordi- 
nary bodies, 

Q  +  /*iVi  .   .  •  +  ^ftVft  —  Igen 

-  -- 


© 


2©      \dvidv        © 

(553) 


2©       Vc?!'!^/       ©  \</     20 

'(554) 
where  (7  =  Q  +  ^^"  ^+  W  "  ^  (555) 

and  A  1/1  =  vi  —  v/,        A  1/2  =  v2  —  v2',        etc.  (556) 

This  is  the  probability  of  the  system  (^  .  .  .  vh).  The  prob- 
abilty  that  the  values  of  v1  ,  .  .  .  vh  lie  within  given  limits  is 
given  by  the  multiple  integral 

*  Strictly  speaking,  rj/gen  is  not  determined  as  function  of  v1}  .  .  .  vh,  except 
for  integral  values  of  these  variables.  Yet  we  may  suppose  it  to  be  deter- 
mined as  a  continuous  function  by  any  suitable  process  of  interpolation. 


SYSTEMS   COMPOSED  OF  MOLECULES. 


205 


20  dvi...dvh. 
(557) 

Tliis  shows  that  the  distribution  of  the  grand  ensemble  with 
respect  to  the  values  of  vv  .  .  .  vh  follows  the  "  law  of  errors  " 
when  i/j',  .  .  .  vh'  are  very  great.  The  value  of  this  integral 
for  the  limits  ±  oo  should  be  unity.  This  gives 


or 


6cMI  =  i, 

ilogi>-|log(2ff®), 


where      D  = 


V 


/  dVn.V 
Wviflfow 


that  is,    D  = 


(558) 


(559) 


'(560) 


(561) 


Now,  by  (553),  we  have  for  the  first  approximation 

H  -  ^gen  =  C  =  1  log  D  -  |  log  (2ir0),  (562) 

and  if  we  divide  by  the  constant  JT,*  to  reduce  these  quanti- 
ties to  the  usual  unit  of  entropy, 

H  -  ^gen  =  ^g  J>  ~  h  log  (27T@) 

.ST  2  JL 

*  See  page  184-186. 


206  SYSTEMS   COMPOSED   OF  MOLECULES. 

This  is  evidently  a  negligible  quantity,  since  K  is  of  the  same 
order  of  magnitude  as  the  number  of  molecules  in  ordinary 
bodies.  It  is  to  be  observed  that  ?7gen  is  here  the  average  in 
the  grand  ensemble,  whereas  the  quantity  which  we  wish  to 
compare  with  H  is  the  average  in  a  petit  ensemble.  But  as  we 
have  seen  that  in  the  case  considered  the  grand  ensemble  would 
appear  to  human  observation  as  a  petit  ensemble,  this  dis- 
tinction may  be  neglected. 

The  differences  therefore,  in  the  case  considered,  between  the 
quantities  which  may  be  represented  by  the  notations  * 

H*en  [grand  »         ^*en  (grand  '        ^^  Ipetit 

are  not  sensible  to  human  faculties.     The  difference 


and  is  therefore  constant,  so  long  as  the  numbers  z>1?  .  .  .  vh 
are  constant.  For  constant  values  of  these  numbers,  therefore, 
it  is  immaterial  whether  we  use  the  average  of  rjgen  or  of  77  for 
entropy,  since  this  only  affects  the  arbitrary  constant  of  in- 
tegration which  is  added  to  entropy.  But  when  the  numbers 
vv  .  .  .  vh  are  varied,  it  is  no  longer  possible  to  use  the  index 
for  specific  phases.  For  the  principle  that  the  entropy  of  any 
body  has  an  arbitrary  additive  constant  is  subject  to  limi- 
tation, when  different  quantities  of  the  same  substance  are 
concerned.  In  this  case,  the  constant  being  determined  for 
one  quantity  of  a  substance,  is  thereby  determined  for  all 
quantities  of  the  same  substance. 

To  fix  our  ideas,  let  us  suppose  that  we  have  two  identical 
fluid  masses  in  contiguous  chambers.  The  entropy  of  the 
whole  is  equal  to  the  sum  of  the  entropies  of  the  parts,  and 
double  that  of  one  part.  Suppose  a  valve  is  now  opened, 
making  a  communication  between  the  chambers.  We  do  not 
regard  this  as  making  any  change  in  the  entropy,  although 
the  masses  of  gas  or  liquid  diffuse  into  one  another,  and  al- 
though the  same  process  of  diffusion  would  increase  the 

*  In  this  paragraph,  for  greater  distinctness,  Hgen|grand  and  %p^lpetit  have 
been  written  for  the  quantities  which  elsewhere  are  denoted  by  H  and  rf. 


SYSTEMS   COMPOSED   OF  MOLECULES.  207 

entropy,  if  the  masses  of  fluid  were  different.  It  is  evident, 
therefore,  that  it  is  equilibrium  with  respect  to  generic  phases, 
and  not  with  respect  to  specific,  with  which  we  have  to  do  in 
the  evaluation  of  entropy,  and  therefore,  that  we  must  use 
the  average  of  H  or  of  7;gen ,  and  not  that  of  77,  as  the  equiva- 
lent of  entropy,  except  in  the  thermodynamics  of  bodies  in 
which  the  number  of  molecules  of  the  various  kinds  is 
constant. 


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